I worked at US summer camps for many years, including into my 40s!!!! My kids now go to one on the East Coast for three weeks every summer and they love it.
There are huge barriers to setting up something similar in the U.K. The main ones being suitable sites and good weather. They need a lot of land and usually have a lake. Most activities are done outdoors, especially where the whole camp comes together. 500 people in the rec hall on a rainy evening isn’t much fun as a one off, so could you imagine the unpredictability of the U.K. weather?
Then add in the need to have a workforce of hundreds, all of whom you only need for 8-10 weeks. You’ll need qualified lifeguards, skilled coaches and people experienced working with children. You can’t just staff it with university students, but older experienced people generally can’t just drop their real job for two months.
My kids camp costs $6k per child for the three weeks (I do get a discount though because I used to work for them). That works out at about £230 per day. I seriously doubt UK parents would want to, or have the means to pay that.
YMCA, Girl Scout and underprivileged camps charge far less / nothing at all, but they are far from that “Parent Trap” experience.