“One of your grandfathers” is one of two specific males (yes, I know it could actually refer to only one, but let’s avoid that bit of scandal).
“One of your great aunts” is one of the sisters of one of your four grandparents.
“Your great aunt” could, informally, still mean one of the (possibly numerous) great aunts you may have, but it could also tell someone that three of your parent’s parents have no sisters and that the remaining one has only one. I believe that you’d not typically infer that, though. Not in the culture I’m in, anyway.
“One of your grandfathers’ sisters” (note the position of the apostrophe) is the same as one of the sisters of one of those two males. Each grandfather may have exactly one sister and we’d still be able to say this.
“One of your grandfather’s sisters” (note the position of the apostrophe) is still one of your great aunts but we may now plausibly infer that one of your grandfathers had more than one sister, and that – if the other grandfather had only one sister – the great aunt in question is one of those rather than the single sister of that other grandfather. But you’re pushing things a bit there.
But in these last two we’ve already missed a possibility. It’s caused by the “One of” bit. It might actually not select a sister at all, as we have assumed here, but a grandfather. It completely alters the meaning. It’s the difference between “one of (his (either grandfather)’s or (this grandfather)’s) sisters” – which refers to one female – and “his (i.e. one of my grandfathers) sisters” – which refers to a whole slew of them.
Without further clarification, it’s not possible to tell which is meant. I’m reasonably sure that the default, natural, interpretation is the single female one. But even there it’s probably more to do with usage than syntax. In typical discourse of that nature you’re simply more likely to be referring to a specific individual.
Were you to continue the sentence and say “One of my grandfathers sisters have formed a choir” (I’ve omitted the grandfatherly apostrophe – it doesn’t elucidate and you can’t hear it anyway) it sounds wrong. You might eventually work it out and realise it does both make sense and is accurate – precisely and concisely informative even – but I suspect you might be a tad annoyed at the speaker for having made you do all that work. In practice you’d be expected to say just “My grandfather’s sisters have formed a choir” and leave open the question (or even relevance) of which grandfather you mean. Or indeed of which sisters – since you still can’t hear the apostrophe, that choir may include all sisters of both grandfathers, but that would be an unlikely intended meaning – again mostly by dint of context rather than syntax.
The question is, is there a language where these ambiguities are removed by syntax alone? Note that I’m not talking about vocabulary and syntax helping out. For instance – and in (very) particular – in Latin we may distinguish a maternal great aunt (matertera magna) from a paternal one (amita magna) but any disambiguation therewith provided is simply an accident of vocabulary. What I mean is, is there a language anywhere (anywhen, even) which forces you – by syntax alone – to be specific so that there’s no doubt that the speaker means “the sister of one of your grandfathers” and not “the sisters of one of your grandfathers” or “one of the sisters of your grandfather” or “one of the sisters of your grandfathers” by synthetic possessive/genitive syntactic marking rather than by lengthy analytic expression?
And would it extend to the disambiguation of the rather large number of possibilities intended by something such as “one of your grandfather’s sisters’ cat’s pyjamas”, where the ‘one-of’ may select (one of) grandfather, sister, cat, or pyjama?