Depends. American Healthcare providers are already here. Once they've taken over, they're then in a position to dictate even more than they are already what care is provided to women in Britain. And healthcare providers can be not just governed by people with more Republican/Religiously polarised leanings over there, they are also rather keen on keeping their major market - the US - with all of its antiwoman agendas happy.
Then there are the pharmaceutical companies - you know, the ones that made literal killings on inflating the price of Insulin and could quite feasibly restrict what medications are available to women.
US influence is very strong almost everywhere. It doesn't have to be sending troops in to take control of far more than a couple of acres in South London and a handful of bases. It's economic, it's data driven, it's data itself.
A random male who fantasises about what he was going to do to an actual neighbour in great detail in himself is of little concern. Once you look beyond the individual to the mass then having influence upon huge structures and organisations, never mind government, then the spectre of the US political giants turning their attention to a piddly little country that's already fucked off 90% of its allies in the last five years, whether for just profit or whether it's also because God has told them it's their job to change it (or their voters have told them that God says it's their job to change it) and phenomenal profit is merely God's earthly rewards and a taste of the true bounty to come - well, it's not exactly something to look forward to.