Nice but not OMG expensive
Unless you move to somewhere right out in zone 5-6*, a 1-bed flat will set you back approx £200+pw unless you want to live in a scummy area. even in scummy areas, you could be paying that and more, believe me. 2-3-4 bed places, you're looking at loads more. Well over a grand, fifteen hundred a month for a 'family' house.
*Could be ok, maybe out West London way, but even then, you're talking a lot for little; London is astonishingly expensive for accommodation mate. And by 'nice' and 'family friendly', I imagine you're talking about something quite suburban, not a council estate.
Plenty of great urban riding around, loads of bike paths and that. West is best for stuff like Surrey Hills, Swinley and that. South is good for that as well, possibly even better.
Be prepared to compromise. Loads. Otherwise you're in for a very nasty shock....