I don't know where you're coming from to compare it to, but London is one of the most expensive cities in the world. I don't know what you call "cheap" but there's nothing in Zone 1 (that is, central London) that will be less than ?1000 a month and I think for a studio it's probably at least ?2000 a month (based on my very cursory searches on sites like Rightmove and Zoopla). For most rentals, you will need a lot of paperwork. They'll want proof that you have a regular income sufficient to pay the rent, for starters.
I have three chums who live and work in London. One struggles to find a place she can live and work (relying primarily on referrals to other working girls on here) and the other two live somewhere cheaper (outside of zone 1, with roommates) and then rent hotels in central London to work.