the UK has basically become the US by most metrics. This includes the increasing privatization of health services, transport, etc... and the excessive commodification of basic necessities like housing.
I find that saying that health initiatives don't work by vaguely gesturing at a country, is not a structurally sound argument. Its like the sentiment here is: "is the fact that we include Pizza as a vegetable in American schools part of the problem? Nooooo, that can't be it. it must be a moral issue!" and thats just one example.
The obesity problem in the US is tied directly to our relationship with highly processed (and CHEAP) food. Along with the stranglehold those companies have over state and federal institutions that allow them to directly sell these foods in schools and institutions, and heavily skirt FDA regulations via lobbying.
The US is uniquely bad when we have a ton of chemicals and ingredients in our foods that are banned in most other countries. It is largely a systemic problem and a problem that can easily be solved. Poorer people tend to eat cheap food, cheap processed food isn't well regulated and is directly tied obesity and a whole host of health problems.