Are you both aware of
IGGY? My DS has not yet done any of their courses - dates have yet to work out - and there seem to be only a couple scheduled (I hope this doesn't mean they're winding down) but we've looked with interest, as it might be a way to get a bit of extra challenge and a chance to meet others a bit like them. (I understand they can be flexible with ages, btw.)
Yes, I am aware of IGGY. (Like you I am an academic.) I did hear that their courses are indeed winding down, though... DD is already doing many online courses with CTY etc and is in a reasonable school situation where we are currently located.
To answer the question of where we would move to: DH and I are academics and we have been approached by a UK university with a view to leading a new research center. There are a number of good private and state schools within the vicinity of the university involved, but none of them has experience with a six year old as advanced as our daughter. I do think we could work with at least one of the private schools in developing a reasonable program for her, but we are definitely going to have to guide them (as indeed we have been doing with her current school).