Fully online GDL and SQE funded by a City firm?

I would expect you to be able to do the PGDL anywhere as it is a standaised course.

For the SQE, many firms do not have bespoke courses or ask that you complete the prep course in London. It will only be those with bespoke SQE in-person courses that it may be a problem, and even then with firms who do have a bespoke course, they can show some level of discretion, around this if you explain it is difficult for you to relocate initially.