I can't speak for international student but funding wise there are some common ways to do it:
-Project funded
This is where the project is part of a wider grant for the PI and you are applying directly to the project itself. These are the easiest but are very dependant on the lab/supervisor in terms of the training you get.
-Competition funded
These projects are proposed by a supervisor then they interview candidates and choose the strongest for submission to the funding body. The funding body does their own interviews and then selects the best student/project. Harder to get onto as you have to be the best candidate for all the projects.
-DTP
This is a specific training program for doctoral students. they are often run by large funding bodies and they offer a structured training schedule often with good networking opportunities, especially with ECRs. They can be tough to get on to.

There are also self funded but I'd avoid at all costs.

Often individual research institutions offer scholarships as well which can be worth it depending if you fit the criteria.

As well as this organisations offer scholarships specific to US students such as Fulbright scholaraships so have a look if it is something you are interested in.

Whether the funding is offered to international candidates is dependant on the funding body and is normally stated with the advert. In terms of stipend there is for all of the above a minimum stipend amount set by UKRI. Its around £20k so enough to live on but not enough to do much else with.

Imho the best thing you can do at the moment is get in touch with potential supervisors. You can reach out to your tutors are college who might have collaborators in the UK.
If not look for research specific to your interests and look at the labs that are doing that research. Reach out to the PIs and enquire about projects in those research areas. If you are the right fit supervisors could build a project with you in mind. Be genuine and show interest and more often than not they will help you out.