I agree with what lx said.
'allophone' is a technical term; the other two are not. However, 'variant' is (to me) more appropriate for describing sounds than 'alternate'-- I don't remember coming across that usage.
By implication, you might find that using 'variant' suggests free variation (non-contrastive variation) simply because you aren't using allophone, possibly suggesting that 'allophone' doesn't apply.
So I am familiar with 'variant' being used for:
1. Cross-variety comparison (individuals, dialects, etc.) where one variety uses X and another variety uses Y. This is especially common in (obviously) sociolinguistic research, but also things like descriptive grammars comparing dialects.
2. A form in free variation with another. English syllable-final Ts have several variants-- aspirated, unrelated, glottalized, etc.
3. I can't say I've come across it used much in historical research even though in theory it would apply. (It might be used to discuss historical dialects, but not in the sense of "the historical variant of modern X was Y". For that I'd expect "form", "sound" or "realization".)
'Allophone' is always used specifically for synchronic phonologically motivated contrastive sounds within a single grammar.