Thanks for all the valuable information. Just looked at the distance places got offered.Everything is well under 20 miles for al the GS that means there is no point of giving even exams because we are in HA3 8QG ( Harrow) which is more than 23miles away .
Move now into the area around the Bucks grammar that you fancy and hope that if your DC fails the 11+, the associated upper school(s) is/are at least no worse than your local comprehensives in Harrow are. I know that a former colleague - an academic - just sent her boys to their catchment school in Wembley, so either she really hates her kids or the local schools aren't necessarily all that bad? Her elder one, being a late August baby, was doomed to failure by the system wherever he went, of course, but that's a whole other thread.
Or don't move, send your child to school locally.
Or put your DC in for the QE / HBS exam and if they get in, either move or let him / her join the traipse across North London and back each day. If you were hoping to get a school place in Bucks, perhaps travel isn't such an issue?
Or go independent?