Post-grad studies are not nearly as competitive as undergrad. You'll still be competing for limited spaces, but against fewer people and with a clearer idea about what you want.
If you had a first from Leicester with really excellent module results, there is no reason why you couldn't get in to Oxford or Cambridge to do an MPhil or MSc.
Even with a good 2:1, the universities you've mentioned like York, Sheffield and even Durham would seriously consider your application.
The fact that Leicester is not a RG uni will make little difference if you've got a good degree. The RG is really just a lobby-group established to promote the interests of a small selection of universities who have high research outputs - basically to petition the government as one collective body on matters over funding, student numbers, fees, etc. It isn't a measure of academic excellence or prestige, though many people have (incorrectly) come to believe it is some objective ranking system of the "best" universities. But the universities themselves established it and control its membership, so it's far from objective. There are lots of highly regarded universities that aren't part of the group, like St. Andrews, Bath and Reading - and Leicester too, which is deemed to be "highly selective" by the Sutton Trust, a much more objective educational charity.