I live near the areas you're looking at and can give you some insight, but if your children are young things at secondary level can change a huge amount before they get to that stage. No-one can tell you how particular schools will be in 10+ years' time.
East Finchley is great for schools, don't know the primaries so well (though Martin is well-regarded) but at secondary you have Archer (very sought-after, has a postcode catchment) and Fortismere on the MH side, also Wren (church school). All extremely popular.
There's a recent thread on here discussing the pros and cons of East Finchley - have a look at the two "where are the nice areas to live in London?" threads down the page.
West Hampstead is (if the Finchleyites will forgive me for saying this) much nicer an area than EF. Fantastic transport links, lovely high street, walkable to the Heath, strong community, great primary schools for all tastes (Beckford - soon to be re-named West Hampstead Primary - is a great all-round local school, Emmanuel is much-loved by the church-going, Kingsgate gets excellent SATs results if that's your main concern). The local secondary is Hampstead, technically just over the border in NW2 but West Hampstead kids get in. It's a good school with a diverse intake (which means people can be snobby about it), but has a new head so will be interesting to see how he does. In the southern bits of WH kids can get into UCL Academy, which is wildly popular if not to all tastes.
The main problem with WH at secondary level is that lots of families in the area send their kids up the hill to the private schools in Hampstead "proper" (WH does not see itself as part of Hampstead), which means people can be dismissive of the local schools. People in the habit of going to church to get a primary place tend to carry on the habit in order to get a place at a church secondary as well, some children end up travelling quite far as a result.
As far as leaving London is concerned - there will always be people who don't want to bring up their kids in the city but equally there are plenty of us who see the benefits of staying. You needn't feel bad if you fall into the latter camp. State schools at secondary level are often better in London in fact than outside - there's been decades of investment to bring them up from the doldrums of the 80s and early 90s.
Please do bear in mind what I said earlier about things changing between now and when your children get to secondary age though!