`Which is correct: a 2-month's leave, a 2 month's leave, 2 months leave. I think it is the first.
two months' leave
Or a two-month leave. We use a singular noun in compound adjectives like this.
It's not right in BrE to say 'take a leave' so two-month leave would not work.
That's interesting. It's natural in American English to take a qualified leave, but I don't think it works to just "take a leave". You could take a leave of two months or a short leave but you would not normally say you were simply taking a leave.