The main State funding in your area is the AHRC, and they now have very limited funding for Masters' degrees. You may get funding for an MRes, which are generally set up to prepare you for a PhD (well, that's the principle I've used in developing a couple of Masters programmes).
You may be able to win funding from the institution in question: either a fees bursary, or some sort of fees + living allowance, but these are increasingly rare.
And generally (well, in the Universities I've taught at) you'll need a First Class UG degree to be in the running for funding. A 2.i is enough to gain you a place, as it shows that intellectually you're capable, but the place and the funding are different things.
You might look at a 4 year PhD programme which is generally an MRes + PhD all rolled up into one, and probably a better bet for funding. But, depending on your specific discipline and university, you'll still need a First to be competitive. The fields of English, History, Theatre, Music, and some languages are highly competitive.
It sounds as though you are going to be able to swim in that pool, so good luck! You need to start looking by December, as most funding applications are in January/February. You'll find information o University wesites, in the Times Higher, and on jobs.ac.uk
You need to be very proactive: you need to contact the Departments where you want to study. There's no UCAS equivalent, and you're expected to do all the groundwork yourself.
Although as we're moving into the 2nd round of the Block Grant Partnerships (BGP2), with universities creating consortia, the landscape is changing a little. If you have very good results, you may find that there are some quite attractive offers out there, although if you're not at a research intensive university it may be more difficult -- I'm afraid I've generally found that my 3rd years (RG, top 10) are better & more on the ball than my MA students who are using the MA to 'step up' from a non-research institution to a research-led course. So a high First will be a very good thing ...