It took us a long time to find a rental that was ok with pets and right for our family. We put in the application, and it took them nearly a month to give us an answer. Turns out the estate agent forgot to mention no pets in the advert. They didn't care about the cat, but were worried about our dog.
I the end they said yes, provided we give them a pet fee and have the carpets professionally cleaned and pet treated when we move out. This in spite of the inspector saying all but two rooms' carpet needs replaced, and the landlords will be renovating when we leave.
We're not in London but rather the southwest. I think it's difficult anywhere these daysthough. Good luck!