F-UBM: Just some idle thoughts. I never was a JW, but I do have a Filipina ex-wife and I was, some time ago, a governor of a church Voluntary-Aided school.
Maybe your wife is relating back to the Philippines where, as you know, religious influence is very strong (mainly RC but also non-conformist protestant). Perhaps you could reassure her that in the UK (or in England, rather) CofE schools are a historical throwback (all mixed up with Church and State, etc.) and if there wasn't a sign outside no-one would know the difference.
Forgive me if I'm out of order but this may be a bit of cultural difference as well as (or more than) the JW thing. House purchase is always difficult (in London many times more so) and IMHO you should stick to your guns. Would it help if you and she visited the school and she could see for herself just how non-religious CofE schools are? I recall having discussions with the head of the CofE school, who had no religious beliefs whatsoever. The parish vicar wasn't sure whether he was a christian, a buddhist, an atheist or all three.
IMHO all religious indoctrination of children is pernicious. CofE schools are perhaps the least pernicious.
Best wishes to you and good luck with it all.