After hanging around Mumsnet for a while I have a few thoughts.
British people don't like living in apartments for various reasons. As the population grows, so to the estates of small houses with small gardens. I'd assume this means less land left over to be simply communal land, for parks, town-adjacent small forests, playgrounds. It also leaves less land for planning anything other than roads for cars. Towns that aren't "concentrated" means everybody wants to drive everywhere, so there's a lot of traffic.
This leads to unsafe road environments where parents aren't happy letting their children walk on their own. Adding to this is a general feeling of unease based feelings of the world being dangerous. Children on their own will be hurt, either by other people or traffic.
There's also not a lot of room left for large schoolyards. At break time children are stuck in quite small areas doing limited activities. They also wear school uniforms which depending on the type might be more or less conducive to activity I suppose (never stopped children in the past who were determined to be active though!).
Parents who try to foster independent children by allowing them to walk by themselves to school and so on are viewed with suspicion and deemed inadequate parents, because they're endangering their children by letting them out unsupervised.
Children are thus confined to their homes and their small gardens. at best there might be a trampoline. Fine for toddlers, kids hitting the age of five and up need a LOT of movement.
www.standard.co.uk/news/how-children-lost-the-right-to-roam-in-four-generations-7264441.html
Instead of obsessing over these things, British people obsess about food and weight as markers of health. What do you eat, is it healthy, is it too much, is there sugar in it, omg what about the carbs. Always weight and dress sizes. Studies show that people who move will be healthier than people who don't move, pretty regardless of weight (unless terribly obese of course, and weight can be a strain on a body in other ways). The focus in Britain doesn't come across as leading active lifestyles though, the focus is on where to park and the evil of bananas (pure sugar).
Previous posters have commented on the excuse of bad weather as well - I live in Sweden which has rotten weather a lot of the time, but you're expected to dress for it. However, when you live in small, damp houses it's much harder to care for all that wet-weather stuff, I appreciate that. In Sweden more and more people are living in cramped conditions, including myself, and just finding space to dry stuff is a hassle. And we USE our central heating the way the lord intended us to. :)
I am of course exaggerating a bit, but this is what I see when browsing around.