Oddly, also in Glasgow, we had no pets at the time we moved. (The last cat had died of old age the prior autumn.) We found a flat and moved in, to find that there were many dogs and at least one cat in the building. We asked the landlord (we are thinking of getting a small animal other than Thor Hamsterdahl once we know we are going to be allowed to stay in-country for sure) and they have no problem with pets.
Our flat has Pergo floors and was completely unfurnished, which helps with the pets, I think. I did view two other places, and one mentioned that they would allow pets with a sizeable deposit without my having asked. (Both unfurnished as well.) Both of the "yes pets" places were of higher monthly rent than the third, and required a sizeable deposit just for the flat, so.... no idea if that is a common thing or not, sorry! And, of course, this probably (whatever is going on here) doesn't apply to where you are looking. But, just the same, it might be that you'll have better luck with unfurnished?