Hey! Current Balliol student here shamelessly wading in to give some (hopefully not *too* biased) advice...!
First, congrats on deciding to apply to Oxford for HistPol! Good choice

I do straight History at Balliol, and I just finished my first year. Can confirm it's great!
I went to a state school and wanted a central, social, friendly and politically active college, too - so pretty much exactly what you're looking for. I'm not an expert on financial stuff, but LMH's post above is spot-on that the Outreach officers will be super willing to help you with that sort of stuff. I also know that financial help is good across all Oxford colleges in a general sense, so hopefully you won't have to make your final college decision based on bursary provisions etc. Also, all Oxford colleges are catered btw
More generally, just some quick info about Wadham and Pembroke first: I don't go to them, so don't claim to know heaps, but defo check their entries in the TSR 'Oxford Colleges Pros and Cons' page (
https://www.thestudentroom.co.uk/wiki/Oxford_College_Pros_and_Cons) - and also check the OUSU Alternative Prospectus (
http://apply.ousu.org). Both of these resources are fab for choosing colleges - honest, up-to-date and a good balance between acknowledging stereotypes and acknowledging that they're not a hard-and-fast rule. It's great you've been to them on the open day and liked them - getting the 'vibe' in person is really the best way to choose. As you know, they're both pretty central (though neither quite as central as Balliol). I wouldn't say Pembroke has a reputation for being particularly political, and it's also not great state school ratio-wise in comparison to Wadham or Balliol - it's only about 54% state school, I believe (tho happy to be corrected on that), whereas in my year at Balliol it's 70-30. It's generally considered a nice college tho, with no stand-out stereotypes apart from being sporty. Wadham is a very politically active college and is known for being *very* left wing. Speaking to people who go there, the general consensus is that that's great if you're really left wing too, and not so great if you're not - as you'd expect haha! Wadham is also known for being very LGBTQ-friendly and hosts a lot of LGBTQ events across the year, so naturally a lot of prospective students who know they'd like to do a lot of that stuff at uni apply there. Both Pembroke and Wadham are pretty social as far as I know.
However, I'm afraid I do really have to pitch Balliol to you!

It's arguably as central as you can get in Oxford, with a Tesco, Sainsbury's and Fever, Balliol's favourite club, literally across the road when you come out of the back door. It's really social and friendly, a large part of this owing to our bar - one of the cheapest college bars and the only one still entirely student-run and student-owned. This means our bar is lively every night of the week, and people from other colleges often come at weekends or on Tuesdays, when all top-shelf spirits are half-price (!!). Balliol is really political, and has one of the most active college JCRs (we do take JCR politics quite seriously!). It's known for being a liberal, progressive college, but whilst most students are left-leaning, there is a healthy appetite for debate and there's usually a range of views on issues. It's really inclusive for state school students - in fact, we're one of only 3 or 4 colleges with a JCR rep specifically for students who've come from state schools/are the first in their family to go to uni/are working class etc. Most importantly of all, perhaps, Balliol has a really strong tradition for History and PPE - we have heaps of History fellows (4, where most other colleges only have 2 or 3), one of our tutors is currently head of the whole History Faculty, and we invented the PPE degree (so it's a centre for Politics teaching). This also means that there are lots of people studying History and its joint schools in each year group - e.g. there are 15 historians in my year, which is exceptionally high. There are also a lot of PPE-ists - people joke that historians and PPE-ists together make up a third of any given year at Balliol, which is great if you want to be amongst like-minded people and love a debate
Hope this helps! Don't worry too much about choosing a college - the three you've narrowed it down to all fit your criteria, so you're in no danger of picking one wildly outside what you want. Obvs get some opinions from Wadham/Pembroke students too, but if you want any more advice/info, feel free to DM me or ask me more qs here!