I grew up in Whitstable: only 5 miles from Canterbury and it's lovely to take the kids to the beach there at weekends..Oyster festival is last week of June.
There is loads of lovely coast to explore with or without kids, e.g. broadstairs, faversham old harbour is lovely, plus lovely countryside near Canterbury, particularly Chilham/Chartham/Wye. It's much nicer than Hampstead Heath! though I did enjoy the swimming pond there. The villages have summer fetes and events with activities for kids, and there are farm type events all the time, which kids love, looking at animals etc. The train to those villages from Canterbury west station is so easy, only 10 mins to Chartham and 20 mins to Wye. Bike rides (with littly on carrier) are great round there as well.
Canterbury is surprisingly ethnically diverse, as there are a lot of foreign students and lecturers in the city (and a hell of a lot of tourists in the summer, which is not so good). There is gentle stuff going on, like art films at the university, exhibitions in Whitstable, etc, but it is not the London panoply of activities.
It is not Camden (I lived there too); it is the provinces, and that can get you down a bit, but the benefits of easy to get to nice places are at least equal to the funkiness of being in North London, and life is a bit more easy-paced, you don't have to be quite so cool, and your kids will have a fresh air and gentle type of childhood..
(can you guess I'm living abroad and homesick by any chance!) anyway, best of luck!