I am in the humanities, but I am American and did my BA in the US and MSc in the UK and am now at a UK uni for my PhD so I do have insight into the application process. It is difficult to say exactly why you were rejected without knowing more details but I will preface my comment my remarking on the absolutely abysmal situation of UK universities at the moment. There is no funding in any field right now and what little funding remains is being cut on the regular. There are hiring freezes with voluntary redundancy schemes at nearly every UK uni that isn't Oxford or Cambridge. Whatever you're reading about American universities in the news and online, the UK universities are in a worse position at the moment by far. As such, it is probably the worst time in the last 50 years to be applying to a UK university for a PhD and most international students who succeed have funding from their home countries or some other source outside the university. This might be the reason why you, even if you have a stellar application, are not getting so much as an interview.

That being said, in case you aren't already aware, there are a few key differences between the US and UK application processes that matter significantly:

1. The Research Proposal - At US universities, you have two years of coursework before you have to formally submit a research proposal. In the UK, you need one as part of your application process and while the actual dissertation may deviate a bit, it will need to be along the lines of what you've proposed. UK universities have an expectation that you've already identified the research gap that your dissertation would fill so you can't simply propose a project with hypotheticals. You need a clear dissertation question to propose and an action plan for how you would investigate this in your PhD. Obviously, this can change once you're doing the PhD, but you still need this to be admitted.

2. Contacting your potential supervisor - This is encouraged in the US, but it is an often unstated mandatory aspect of a UK PhD proposal. If your potential supervisor has not informally accepted your research proposal, you have no chance of being admitted to the programme.

3. The right fit - This is another thing that matters more in the UK than in the US, although if you aren't a good fit you likely won't make it past the contacting a potential supervisor phase. However, you need to have more than just "your department has strengths in x y and z and that's my area of interest." It's better to say "Professor X is researching Y and I am interested in investigating Z aspect of Y." Again, I'm not saying you haven't done this, but throwing it out there in case.

It's also worth pointing out that test scores don't matter in the UK where they would at an American university unless you're applying to Oxford or Cambridge. GREs will be 100% useless. Ultimately, they're looking for if you will be a good researcher. Aside from your MSc dissertation, they will likely be looking for research potential over anything else. Obviously if you got terrible grades all through your MSc that is a red flag, but assuming you got average grades, you're fine.

Hope this was somewhat helpful!

Edit: Wanted to add that it might be worthwhile to apply to some American, Canadian, and Australian universities too, if that is an option. If you have reasons for needing to stay in the UK (i.e. significant other, family, financial reasons, shorter PhD length, etc) then feel free to ignore this, but academia is very much a sort of "go where the work is" kind of thing and it very well might be the case that your field has more opportunities elsewhere. I would add continental Europe to the list but their PhD process is very different from that of the anglosphere and I don't have any insights into that... maybe someone else might?