Why should someone who is insane be compared to one who gathers nuts?  Is it from the expression “gathering nuts in May”?

This use of “nutter” post-dates, and presumably derives from, “nuts” [edit: or “nutty"] to mean “insane”, the origin of which is discussed here.

I suspect that “nutter” in the sense of “one who is nuts” was coined independently of the sense “one who gathers nuts”.

I think that nutter is derivative in much the same way that “looney” is from lunatic (which, by the way, has a rather interesting pedigree all its own).

I think I’ll put my money on it coming from the sense of ‘off his head’, as the use of ‘nut’ for head appears to predate the off his nut usage by about a decade in the mid 19thC.