Before you apply for any scholarships or minis, buy 'Bewigged and Bewildered' and 'The Path to Pupillage' (which has a new edition about to come out) - both excellent guides to the GDL, BPTC, scholarships, mini-pupillages etc. They demystify the process, particularly if you haven't done a law degree.
With an Oxbridge first you stand a good chance of getting mini-pupillages, even without other legal work experience. Minis are intended to give you a taste of the bar, and chambers don't necessarily expect you to have started the GDL already (although some chambers do expect this - have a look at a few websites to see who the minis are aimed at).
Send off a few mini-pupillage applications, and see how you do. If you find that you don't have much success and people tell you to come back when you've started the GDL, I'd recommend just spending a few days in the Old Bailey, local County Court or Royal Courts of Justice listening to some cases and seeing advocates. This will be something you can talk about on GDL scholarship and GDL application forms, and will give you more of an idea of the day to day life of a barrister.
As an aside, I probably wouldn't do minis at your 'dream sets' now - save them for when you have more experience and can impress chambers a bit more. Also, try to apply for minis at a range of sets. The criminal bar is difficult financially but some people can't imagine doing anything else. 'Human rights' or public law pupillages don't really exist in isolation - chambers generally do elements of human rights alongside criminal / immigration / even commercial law. Try some civil sets too - you might be surprised how much you enjoy areas of law such as Personal Injury / property / commercial, and it is a much more financially settled career at the moment.
GDL scholarships are not as valuable as those for the BPTC. You stand a reasonable chance of getting one, as long as you can put together an application which is clear about why you want to become a barrister, but it is only likely to be in the region of £5,000 - £10,000 maximum. This will probably cover most of the course fees outside London, but you'll have to fund your own living expenses.
Finally, you definitely don't need to move to London at the moment. The City Law School GDL has the best reputation for barristers, mainly because more people on it are aspiring barristers than is usual on GDL courses, and it is perceived to be more academic (though plenty of successful barristers and pupils didn't do it and it hasn't affected their chances). However, it's not worth getting into significant debt to move to London if there is another course nearer home. The BPTC is a different issue; personally I think you have a better experience in London because you can be more involved with the Inns and are closer to chambers for mini-pupillages etc, but that's a whole different debate!
You're still young to be thinking about the bar. Plenty of people don't do pupillage until they are in their mid to late twenties. It's not the only great career out there, but if you think it might be something you'd enjoy, it's definitely worth going for it with your academic background. Good luck!