I'm so sorry to hear that your school is being unhelpful, because the standard way to reapply through UCAS is still via your school. And having just finished written what you're about to read, your school is doing you a massive disservice by not helping you. But let's try anyway:
You should clarify with your school where they stand. They're unhelpful and telling you not to resit but if you tell them you're going to ignore them, will they let you resit your exams there anyway? If not, you will need to find a private exam centre to resit.
In terms of filling UCAS, you'd do exactly as you did this year. You need to declare your achieved A-level grades, and provide a reference, which again, would typically be your school - they might be inclined to provide you a reference still. Your referee is the one that provides the predicted grades, so that's a discussion you'll need to have with your school or, whomever has written your reference. Anecdotally, I've heard of a handful of people having references written by private tutors, so this is something you could look into if your school continues being un-cooperative, but that comes with its set of challenges too and ideally, you'd want to avoid this situation in your position.
Also, I would look beyond London Russell Group universities. It's fine to have preferences, but at the end of the day, the Russell Group is nothing but a clever bit of marketing by a bunch of universities. There is much more to a university than perceived prestige and its location and you might be happy in other places and just as successful if you're so inclined (the numbers show that it's the grade you get in your degree that matters, not the uni you went to). At the very least consider and apply to a handful of other universities, because it's risky business putting all your eggs in one basket. And are you sure you're actually uninterested in your insurance or are you just writing it off because it isn't a Russell Group? There really isn't any shame in going to a non-Russell Group uni or a foundation year - literal thousands of people do this, and so it's worth thinking your insurance through a little more, particularly considering that your school is not very willing to help you and therefore making things very difficult for you. So I would continue reconsidering your insurance uni as an option for now.
I'm a little concerned by the fact that you're asking how you should study on your own. You should have a good idea on how to do this by now, but oh well. The usual thing to do is to plan how you're going to study (everybody has different methods because what works for me might not work for you) and set yourself a revision schedule, and then actually stick to it. However, the fact that you're asking makes me thing you'd probably benefit from spending a little more time in a structured learning environment, like school, so I'd ask your school if you can redo year 13 there. Since it looks like your school might not be the biggest fan of that, I'd also call other sixth forms in your local area as a matter of urgency to ask them if they'd let you redo year 13 there. And failing that, I'd try a private tutor to get you started with some sort of structure to your studying.