We're not British and recently moved to London from an affluent small city elsewhere in the UK.
Pros:
More activities for the children in SPADES - sports, creative arts, languages etc. It's beyond compare.
More compressed neighbourhoods = more local friends for children.
Better sports facilities, locally. In our old city, the sports centres were outside the ring road and impossible for us to access in under an hour without a car. In London we are within walking distance of about 3 swimming pools, one of which is less than 5 mins from us, never mind all the other sports.
More choice of schools - in our old city we were in the catchment of one school. Where we are in London (zone 2), about 8 primaries and maybe 3 or 4 secondaries. Obv the most desirable have tiny catchments though.
Better, cheaper public transport. The tube is £££ but the bus & overground are cheap.
Better entertainment for us as adults. It's barely more expensive than the massively expensive city we moved from.
More babysitters. Infinite babysitters! It was a nightmare before.
Waaaay more to do at the weekends and half term. We can walk along the canal to fab street markets, cafes, exhibitions etc. With a couple of well chosen memberships like the Royal Academy or the V&A you have free exhibitions all year. We're 25 mins from the Natural History Museum. Half terms are a positive JOY compared to the boredom in our old city where once you had done the one museum that was kind of it. Here in London you can pack every day with brilliant, diverse fun.
It doesn't take a million hours and £££ to get to the airport.
Jobs have higher pay and there are more jobs, with greater diversity of choice etc (i.e. if you want flexible working etc), and commutes for those high salaries are shorter (unless you are unlucky).
Cons:
Property prices/rent.