If you're doing single honours English then you could use some elements from another discipline in your dissertation: a particular exercise in historical method, or a way of handling bibliographical evidence borrowed from archaeology, or the thought of a particular philosopher, say—any of those might in the right circumstances be an appropriate tool to turn to. When I was an undergraduate about a quarter of my dissertation was a mixture of historical underpinning and some art-historical readings of a set of sculptures (about half was a literary-critical argument from close readings, and the other quarter was the introduction and conclusion). You'd probably need to justify your choices, at least briefly, in your introduction, and possibly you'd want to explain some of the workings of whatever tool you were using so that your markers could be comfortable with it. The dissertation would also still need to be recognisably work in the discipline of English, though if you think about what people who work in English departments publish that's a pretty wide range.
As I understand it if you do a joint honours degree course (e.g. Classics and English or History and English &c) you've more room to do a fully interdisciplinary dissertation.
Reading widely, and well beyond the scope of your A level (or equivalent qualification) is definitely an advantage, and reading some Middle English is one good way to do that. In secondary education most of the texts students encounter can often be within a few narrow categories: often, short poems from the Romantics or from more recent poets, nineteenth-century novels, modern drama, Shakespeare and possibly one or two playwrights contemporary with him. So try to read at least some things that aren't those! Challenge yourself to travel widely across times and genres.
If you're interested, there's a big library of free Middle English texts online
here, with introductions and marginal glosses explaining unclear words. The texts vary in difficulty but many can be read fairly easily by modern English speakers (and the first thing to do if a sentence is giving you trouble is to say it out loud to yourself: this often clarifies it). You'll also find that you speed up quite quickly as you get used to it, if you persist.
Here, to pick an example more or less at random, is one medieval version of the myth of Orpheus.
The various pages
here contain some useful advice about preparing for applying to Oxford, and most of that advice would be useful in applying to Cambridge. And indeed a lot of the notes on preparation on
this page will stand you in good stead studying English at any university. (This website is not my work!)