Just seen this so apologies if you have already made a decision. Bromley North is not exactly a no go area exactly for primaries but it can be very tight for certain schools. If you haven't bought yet, take a look on the LA website (www.bromley.gov.uk) at recent distances for some of the more popular schools, e.g. Valley and Parish particularly and try to buy as close to the school gates as you can for either of those.
Although Parish is C of E, I think its criteria is purely distance related. The only Bromley North school that I know involves religious criteria is St Josephs which is Roman Catholic and is single form entry so religious attendance is likely to be an issue in most years I think.
Scotts Park school is a community school (so no religious criteria) that is also expanding to be a permanent three-form entry, so it should be a safe bet (as far as these things can be) for people in Bromley North. It is in a state of flux at the moment as it had a poor OFSTED last year but I believe it may have a temporary head at the moment and it will be keen to turn itself around if there have been problems so is likely to be a good bet in years to come.
These are the schools actually in Bromley North but there are of course others which are driveable from there - a new Harris Shortlands which is temporarily based at Bromley South but will be moving, a new bilingual French-English free school, La Fontaine Academy, which is temporarily based at Bromley Common by Princes Plain Primary School, but again will be moving though I don't think there is a definite location yet.
So quite a lot of options in the area but you do need to do some research before you pick a house.
Good luck!