Some time ago, when I was living in Oxford but technically registered for a London PhD, the funding authorities agreed (in writing) to pay me at the “London” rate of grant.
As many Finalists, Cambridge Part III students and taught Master’s course students will already be aware, the finding situation in higher education is very tight, and as a consequence, not everyone at Oxford/Cambridge who wishes to stay on for an Oxford D.Phil or Cambridge PhD will be able to do so. Those Oxford/Cambridge students who are unable to secure Oxford/Cambridge grants may wish to consider the possibility of continuing to live in Oxford or Cambridge privately and registering for a PhD at one of the many London colleges.
Birkbeck College, SOAS, and UCL are within easy walking distance of Kings Cross British Rail station. QMW is only a few stops away from Liverpool Street British Rail station by tube. There are also many other London colleges.
Some London University academics themselves live in Oxford/Cambridge and commute into London. These would make ideal supervisors for a student who was living in Oxford/Cambridge but registered at London, as supervisions could be held in Oxford/Cambridge. These academics can be identified by cross-checking the Oxford/Cambridge telephone directories against the London University faculty lists.