This is an interesting question. It is important to remember that the IE family is well-established because there is so much data available to us, and especially because we have historical records of classical languages. It was originally similarity between Latin and Ancient Greek compared to Sanskrit that suggested a relationship, not a similarity between Italian and Hindi (or Irish and Punjabi) or other modern languages.
The relationship between the classical IE languages is often quite clear, and the time depth there is around 3,000-4,000 years, versus 6,000+ for the modern languages.
Having just two languages also makes the comparison harder because we only have two data points. If the two languages happen to vary by chance (e.g. borrowing in one or both languages), we wouldn't have any way to observe the original similarity. If we compared all of the Celtic languages to all of the Indo-Iranian languages, for example, we'd have a lot more data to work with, and more chances to observe similarity. But having even more branches means we get to try to understand the full story, and how Irish and Punjabi each fit into it, rather than just comparing them individually with a lot of historical gaps.
But to take an extreme example for comparison, Basque has been extensively studied to look for possible relationships with just about every other language out there that might be even remotely plausible. (None of those possibilities have worked out.) So if Irish and Punjabi held the same status as linguistic isolates today, presumably someone would have tried to compare the two (especially because they would
both be isolates, so the mystery for both would be interesting, and suggest a possible relationship, or at least both would interest people who wanted to study isolates). So I don't doubt that someone would actually try this study. That is, assuming historical linguistics progressed as well as it did based on Indo-European, and the same questions were being asked today. But I'd think that with Austronesian, for example, where the reconstruction is more transparent and in some ways easier than Indo-European, the scientific methods would have eventually ended up where they are today although it might have taken longer, because the original interest in IE was partly due to nationalism, race, etc. (which, tangentially, was part of what fueled the World Wars, Nazism, etc., and some of those ideas of nationalism in connection to, e.g. "who really belongs here", etc., still persist today, including in relation to where the IE homeland was, etc.). So let's set all of that aside and assume similar methods in reconstruction, and also that someone attempted this study.
In terms of the difficulty of showing a relationship, Peter (1991, an honors undergraduate thesis at the University of Illinois at Urbana-Champaign) showed that if we compare just modern English and Hindi, looking for apparent "cognates" in the languages, only about 35% of those would actually reflect a common origin. I'll assume it's similar to Irish and Punjabi (maybe somewhat more, given the extensive borrowing in English [from French] that would make direct reconstruction harder, so maybe 50% of apparent cognates would be actual cognates). What this means is that we need a reliable method to separate out unreliable data from real cognates. But that's the foundation of Historical Linguistics. If you look at any two languages, you can find coincidentally similar words. That's not enough to show a relationship. Instead, what you need to find are
patterns. And specifically, you don't want to find identical things in two languages, but
systematic differences. Consider Grimm's Law for Germanic: you would find a correspondence between Germanic /f/ and /p/ in other languages (e.g. Romance), and so forth. You aren't looking just for simple equivalence as in /p/=/p/, but evidence of
regular sound change producing systematic, corresponding differences like /f/ and /p/.
As a basic starting point, we need data, and many linguists would start with the Swadesh list or something like it. We happen to have those available for Irish and Punjabi:
https://en.wiktionary.org/wiki/Appendix:Irish_Swadesh_listhttps://en.wiktionary.org/wiki/Appendix:Punjabi_Swadesh_listNow if you just skim through those lists, they look quite different, and it's not immediately apparent that the languages are related. But we can use the sound patterns to look for possible connections. I should emphasize that I admit my perspective here is somewhat biased by knowing about PIE reconstruction already, so the hypothetical linguist in our scenario wouldn't have that same information. But it's really not all that different from other groups that have been reconstructed.
Let's look at a few basic words, which happen to often be similar in related languages:
English: Irish / PunjabiI: mé / maĩ
you: tú / tū̃
he: sé / iha
(Caveat: we'll rely on these forms even if they are likely somewhat obscured by orthography, especially for Irish. [Some IPA transcriptions are given there, but not all.] Regardless, that's not a practical problem for serious research and for linguists familiar with the languages, just a limitation of my answer here based on that source.)
These pronouns are far from conclusive, but the first two certainly look similar. Assuming the languages are related, then either /m/ and /t/ are inherited directly with no sound changes, or they changed in one or both languages but ended up converging again. As for "he", there's no obvious connection there, although we might wonder about some kind of relationship between /s/ and /h/, which is attested in some languages (e.g. Coastal Latin American Spanish at the end of a word, where
los niños may be pronounced as
lo(h) niño(h)). Regardless, we need more data.
Looking at another set of words that are commonly shared:
English: Irish / Punjabione: aon / ikka
two: dó / do
three: trí / tinn
four: ceathair / cār
five: cúig / pañj
Some of those have obvious apparent correspondences, and others do not. "One" is quite variable across languages, so that's not surprising. You would think that "four" might show a relationship with /c...r/, but then "five" is either a different root (borrowing or other lexical change) or suggests instead /c/ = /p/. So this is starting to get complicated. Following up from about about /t/, here we see /t/=/tr/, so we would want to try to understand what's going on there. Maybe the sequence /tr/ was original, and either /r/ in general disappeared in Punjabi or in the cluster /tr/. But above we know that original /t/ probably corresponds directly to /t/ in both languages, so it's unlikely that, e.g. /t/ became /tr/ in Irish, unless of course it has something to do with the adjacent vowels or something else. Or maybe these /t/ phonemes actually represent different original phonemes in the proto-language.
To summarize at this point, even with very little data we've seen some hints of possible relationship, but so far nothing conclusive. We would want to gather some additional hypotheses about possible correspondences. We can do that just skimming through the list. Again, here it would be especially helpful to find possible correspondences between
different sounds, and see if we can verify those as general patterns. Today, we might attempt this with a computer: look at all possible correspondences, and see if some seem to appear repeatedly. Or we might try it by hand, following traditional research. Regardless, this is just guessing to see if anything happens to fit.
The results would be mostly noise, to be thrown out. But if we do stumble upon a genuine correspondence, and then observe that correspondence in many more words, that is worth pursuing. If we can find enough such correspondences, we may be able to build up a systematic reconstruction of the two languages back to the proto-language and list the sound changes that occurred in the development of each. Just glancing through these lists, it is immediately apparent that there are a lot of borrowings (or other kinds of lexical change) obscuring any possible relationships. Syllable structures are different, etc. But if we keep looking at the data and trying out various hypotheses of correspondence, we may find something.
So a couple more pairs to consider:
English: Irish / Punjabiand: agus / atē
to see: feic /vēkhṇā
mother: máthair / mātā
The first pair suggests /g/ = /t/. That happens to be incorrect (I know this for reasons our hypothetical researcher would not, based on the etymology of the words), but we could test it with more words. There's no reason not to test that. Do we see that correspondence often? If not, we can reject it as coincidence.
The second pair would suggest both /f/=/v/ and /c/=/kh/. I believe the first doesn't really work out (at least not as a general correspondence, because it was a somewhat unusual change in Irish), but the second probably does.
And the third pair suggests "th" [h] =/t/, and these words are in fact historically related, so looking at more might reveal this as a pattern. But this is actually misleading: the Punjabi word for mother was actually borrowed from Sanskrit. So reconstructing this would lead us in the wrong direction, although it might also appear in other words borrowed from Sanskrit.
If we repeat that process enough times, we might be able to separate out the noise from the genuine correspondences.
But at this point one thing does become clear: we need more than the 100-200 words of the Swadesh list, and these are generally thought to be among the most likely words to be cognates, so we would have even sparser results with longer lists, but we could still find some. A fundamental problem here is due to borrowings, so it's not entirely clear to me whether this pair really is reconstructable. It might be similar to many cases we see today of proposed relationships that cannot be sufficiently demonstrated.
In the time it's taken me to write up this answer, I've failed to sufficiently demonstrate a relationship, and that's even with the help of knowing something about the relationship between these languages (although for the most part I was guided by the data itself, i.e. not looking up the 'answer' to this question directly).
In terms of systematic correspondences, though, I do see one area in the data that looks promising. The wh-question words and the demonstratives seem to possibly show some correspondences. The Irish wh-words begin with "c" (=/c~k/) corresponding to /k/ in Punjabi. Of course those probably aren't independent data points (all going back to one root), but the repeated correspondences are what we're looking for in order to have some confidence in the correspondences we find. Similarly, I see Irish "s" (=/ʃ/) and Punjabi /h/ or /th/ in demonstratives or related words like 'this', 'here', etc., so I'd check that in more pairs.
At this point, I'd think the next step would be one of the following:
1. Learn each of these languages well (or spend a lot of time looking at dictionaries) to see what additional pairs can be identified. It would be very helpful to remove/identify any non-root parts of words (e.g. inflectional suffixes, as well as compounds, etc.) if possible, and a speaker of each language could do that more easily than I can. That might remove some of what's obscuring the relationships. Similarly, rather than relying just one the one-item Swadesh list pairs, we could ask speakers to name various synonyms or related meanings for each item to see if there are some other words that do show correspondences not demonstrated by these most basic translations from English.
2. Try a computational approach, that simply looks through every possible correspondence then lists out those that seem to repeat fairly often in the data.
3. If we had any information about borrowings and neighboring languages, we could try to remove some of that from the data. Your question suggested that English and Sanskrit don't exist, or at least there is no record of them, but in a real scenario of this type we'd probably know something about some of the neighboring languages, and even if not related they might be a source for borrowings. In the specific scenario that Indo-European developed exactly as it did but then mysteriously the only remaining languages today are Irish and Punjabi with no trace of the others, that would make this extremely difficult.
In short, this is very difficult, but not impossible. Persistence in research would suggest that eventually someone would stumble upon some reliable patterns in the data, and we might be able to sort out the rest from there. But having only these two data points makes it very difficult to understand anything like the full story we know about Indo-European today.
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...both have developed features that are unusual in IE languages - in the case of Irish, initial consonant mutation and in the case of Punjabi, phonemic tones.
Those features are not, in themselves, dealbreakers here. Tones came from somewhere, so we could possibly reconstruct that (I'm assuming some kind of coda consonant), but vowels aren't that reliable anyway for reconstruction, so maybe we just set them aside. The initial consonant mutations make the reconstruction trickier, but just give us more options for correspondences, and instead we could focus on non-initial sounds. Together, it seems like mostly medial sounds might be most transparent. But my guess is that these features don't make this pairing much more difficult than, for example English and Hindi, or other similar pairs. (There are some pairings that would be easier, though, especially any languages that don't have as many borrowings as both of these. I wonder if Swedish and Russian, for example, would work out better.)
