Which masters program at Oxford do you want to enter? Each of the Oxford English MSt degrees and the Mphil state that they require: a first-class or strong upper second-class undergraduate degree with honours in English literature and/or English language, or exceptionally a related subject. So, you already have your first. Now, you will need three excellent recommendation letters. Don't underestimate this important component of your application. Regarding your dissertation, were you planning on asking for a reference letter from someone on your dissertation review committee? If so, I think you should make the effort to discuss your paper with that person and get feedback on ways to improve. Share your thoughts on pursuing a graduate degree and get feedback on how they think you might do in a graduate program at Oxford. If you get any negative pushback, find someone else to write your recommendation. Ideally someone who knows your work very well and can attest to your ability to excel in the masters program.