I didn't realise students couldn't apply direct to research councils for PhD funding. I thought that was just for Masters studentships.
Apart from research councils, the other sources will depend on your project area. For example, the British Heart Foundation accepts applications (I think) all year around and will take applications from students/their supervisors, but that is only relevant if your project has a cardiovascular angle. So try to identify disease-specific charities which might support your work.
If your project has a potentially industrial component, would a CASE studentship be something your supervisor could try to get hold of, in collaboration with an industrial partner?