I'm in Redbridge but am not planning on staying here - hoping to move.
As other posters have suggested, it would be best if you could say which area you are planning to move to. Are you planning to stay north of the river or to move somewhere in south London? From what I know, north London schools' exams are somewhat different from those in south London/Surrey schools. If you could tell us which area you would want to move to, we would be able to offer you some advice relevant to your circumstances.
To answer some of your questions:
You can apply to any grammar school you want, although some have catchment areas, so you need to find out about those beforehand. I would also suggest that you take into account how long your child's journey would be to each of the schools you're considering. Please don't opt for a school which would involve a lengthy and tiring commute.
As for your 'can a child sit an exam anywhere?' the answer is: not really. It depends on which schools you are going to apply for, as they all have different exams arrangements; some have exams of their own and some get together and have a joint exam to cover all the schools in the group.
Finally, if you have a bright and motivated child (as you indicate you do), I think he can do it, but you have to get him to read as much as possible and in various genres, starting now...
(ETA: cross-posted with Guest55)