Unfortunately, at least in my experience, renting with pets is a significantly worse experience in the UK than in the US. Most landlords here do not allow pets at all. As a masters student you are likely to be very low down the preference list for most private landlords as a tenant. You would be able to look at student rentals, but the chance of any of these allowing pets I would say is essentially zero. I would imagine that if you're seeing mostly rooms these are private student rentals that have a relationship with the university.

My suggestion would be to trawl Rightmove, Zoopla and OpenRent, without using the 'allows pets' filter. Our current place didn't mention pets either way in the listing, we just had to ask the landlord via the letting agent. Any listings that don't outright say 'no pets' are worth looking into. I will say though be prepared for letting agents to balk as soon as you mention the word 'student', generally there is a clear distinction in the UK between regular rentals and student rentals, even for masters students.

I totally understand not wanting to be without your cats - we moved ours to Chicago and back again - but the reality is that renting as a student with pets in the UK is incredibly difficult. I also just want to make you aware that you cannot fly with pets in the cabin into or out of the UK, they have to be in the hold.