PhD funding source for international student in the UK

Your best bet is probably to check if there are fellowships available for study abroad from your home country. However, depending on where you're from, there may be special programs available. For instance, there are Commonwealth Scholarships available for people living in members of the British Commonwealth.

There are some databases of UK awards for general students: here, for instance, is a list of postgraduate scholarships available to citizens of any country.

The UK PhD student funding situation is quite dire, due to the concentration of funding into Centres for Doctoral Training (CDTs), a special type of institutional grant from the Research Councils. It means that there is no funding available in normal project grants to hire PhD students. If the university/department doesn't have a big CDT grant in your area, you're pretty much out of luck---as is your professor/lecturer in terms of being able to expand or in case of young lecturers even build a research group. There's only very limited funding based on REF results, which are then fixed for several years, and there is a brutal competition within the department for those studentships. (My department has more lecturers/professors than PhD studentships.) The next REF isn't due in many years. As already mentioned, your best bet is a grant from your own country. Indeed, students with their own grants from their home countries are the best bet for many UK lecturers to get any PhD student at all!

You may also use website that are specifically targeted at helping at finding PhDs. The following website allows you to filter using funding and nationality as parameters: http://www.findaphd.com/