@TheGoldRobin We moved out from London and while each grammar and secondary has specific catchments, found many children our kids' primary school were able to get places outside of the specific catchment. That varies year to year so it could be one year is harder than the other...
We looked at houses and schools across parts of Chesham and Amersham to Wycombe. We chose a suburb between Amersham and High Wycombe (Holmer Green) - bit more space for us and the option to take bus or drive to Amersham,/Chesham tube stations as well as High Wycombe rail station.
As luck would have it, children in our primary were eligble, at the time, to apply to the High Wycombe, Beaconsfield, Amersham and Chesham Grammar options.
If I would do it differently today, I would probably focus less on primary catchment and more on secondary - from the folks we know, primary schools are generally pretty good in this area. Secondary schools vary a bit more, but if you dont find grammar places upto GCSE, there is always another chance for A-levels.
It wasn't a huge number but we were aware of in-year transfers in year 8 and 9 for a number of the grammars. As others have suggested, it's best to talk to the specific school but often they will only really engage once you have moved into the area. Depends on the intake for that year - people are always moving, transferring etc...
Your school options, in my opinion, are good. Pick the area with the house/street you like and that works for your family.