A lot of folks have successfully done it, but it usually takes at least a little bit (if not a lot!) of extra work. We brought our 2 cats and have successfully rented 2 flats in London. We looked for flats without carpet, so hardwood, laminate or tile floors only. We looked for sparsely furnished apartments, so there was a low risk of a landlord envisioning ruined upholstery, we had a "pet reference" from our previous landlord in the US and, for the first apartment, we offered an additional 2 weeks of deposit.
Our current place didn't really care at all.
We did run into a decent number of places that were just flat out "no pets" due to overarching building policy -- didn't even get to the point of dealing with an individual landlord. We were really open about our cats, so I was always annoyed with agents who showed us places with no-pet policies.
Good luck!