Hi there,
I'd put forward the suggestion that all cities, towns and to an extent, villages have accessible public transport. Of course, the services in the cities are going to be more frequent but around local small towns and villages there's a more than adequate service/provision.
I think it's only once you move out into the countryside that you really start to 'need' a car in comparision to public transport services.
It's also impossible to tell or compartmentalise 'do you need a car if you have small children' - I know and have seen plenty of families and single mums etc who don't have a car and seem to be functioning very well, others who have small older runabout cars and yet others with the latest BMW's and Porsche SUV's - it really comes down to your individual requirements and lifestyle - a car more than likely won't 'just' be used to ferry the kids around and buy groceries, you'll probably want to use it for socialising, visiting and holidays, possibly commuting, and maybe leisure activities etc - you need to look at the complete potential scenario you'll be in.
As already mentioned, you can pick up an older decent runabout for peanut money (I've been to car auctions and seen road legal cars sold off to £25 ! - they do need work of course, but hey, had some road tax left and some petrol and with an MOT certificate!) but usually 'cheapo' cars will set you back at least £200-500 for which you can get a perfectly useable car.
which area or city, town, village are you considering to live in?
Cheers, DtM! West London & Slough UK!