We had a terrible time in Bristol finding a place that would take a cat. Estate agents actually told us they wouldn't bother showing us any apartments, because they knew the landlords would not take pets. Reflection of the housing market, I suppose. When there are plenty of tenants and few spaces, landlords can be picky.
At any rate, in the end we found a place we really loved, and just kept our mouths shut about the cat until we actually had seen the place, established a negotiating relationship with the landlord, and taken a look at the lease. It had no pet clause and in the end it turned out the landlord didn't care. But I think we got lucky.
One thing I've noticed about UK flats is that many of them seem to have carpeting instead of hardwood or laminate floors. I was a bit surprised at this, but it may help to explain why no one wanted a pet. I'm not sure if it's the case in other cities, but you might want to give yourself a big head start on searching, just in case.