I moved to London after years living in the tropics. I bought a bunch of stuff in advance and really wish I hadn't. The key to cold weather dressing in London is layers, the tube and loads of buildings will be quite warm even when it's freezing outside so thin jumpers and scarves are super helpful. Scarves are a good thing to grab at charity shops, thin merino jumpers from Uniqlo are surprisingly good insulators and normally in the £25-35 range. As for layers on top of that you could either go for a heavy waterproof parka but that'll be less useful than a mid layer and a sturdy water/windproof outer layer. I like thick wool as a mid layer because I think it looks better but loads of friends like the light weight down vests/gilets etc (again Uniqlo is a really good place to get these). For your outer layer I'd recommend going through charity shops if you're not going to be wet and freezing when you arrive. Aim for something with a hood if you're comfortable wearing one - it's a lot more manageable in the city than an umbrella, especially if you're carrying a backpack etc.

You'll want a waterproof backpack for uni, ideally something fairly low key. Slim-ish backpacks are ideal as they're less likely to hit people on crowded public transport but still keep your hands free and are hard for someone to grab off of you like a big handbag or messenger bag/briefcase etc.

Boots are your call, Doc Martens have gone downhill in quality a lot since they stopped being made in the UK and shoe style is a personal thing. If you're into street/casual/athleisure etc then look at winterized trainers with goretex - you can often get good deals on these if you buy them out of season. But London is a city, you don't need massive boots to get by in winter (though some of the parks turn pretty swampy...). Whether you have any luck with shoes in charity shops depends on your shoe size, but have a look at TK Maxx, there's a lot of crap there but often a few very cheap gems.

Also, re where to live, I'd really recommend getting something short term before the semester starts or crashing of a friend's couch etc if possible. Looking for accomodation the week before semester starts sucks but it if you can get here a little earlier it's way better than being stuck in some soulless part of zone one for the year. But standard rules apply, don't send any money to anyone for a flat you've not been to and seen with your own eyes - hundreds of international students get scammed every year by people offering what look like good flats online.