I've been to two and had very different experiences.
One I didn't like at all. Too many cats, not enough staff keeping an eye on people mixing with the cats and the cats didn't have anywhere to escape too.
Second was fab. One room with tables and chairs where you have tea/coffee and cake. They don't make food on site, they bring in cakes and biscuits made by another local company.
There was a list of rules about interacting with the cats - including no lifting because the girl in charge pointed out lots of people lift cats who don't want lifted, or don't lift/put them down properly.
Then a huge living room type room with sofas, cat climbers and toys that you went into. There was two doors off it with cat flaps - no visitors were allowed through that, or to block them, one goes to a room next door and one goes upstairs where there are 2 or 3 rooms. If the cats wanted to fuck off somewhere quiet then they could. Some did, some came in and out and quite a few others jumped up on laps and got petted.
The owners make it very, very clear that you are paying for tea/coffee and cake and the chance to spend time with cats. If all the cats don't want to play then that's tough.
They have a wall of cats from the local rescue place advertising cats who need homes.