Since there are hundreds of universities in the US and UK, you should start with Google. Research which universities in each country offer a PhD in the discipline in which you're interested. Once you've compiled a list of schools, start researching which of them offer scholarships and teaching assistantships to their PhD candidates. If you're lucky, someone might have already compiled this information online, for your particular PhD track.
You'll have to do this quickly, because most applications and support materials are due by February for fall admission.