I've been to the bar/restaurant in City Screen a few times to meet a friend. It's a great location, busy without being too busy to get a seat and most importantly - spacious. I don't take my pram to York (can't get on the bus there with the pram), but the friend I meet takes hers so meeting somewhere with space is essential.
Staff are polite and helpful, it wasn't a problem to get some hot water to warm Missy's bottle up.
It's a slight pain to have to go to a different floor for baby changing - and also get the key from the staff - but the plus to that is the facilities are nice and clean. It's no big deal to not have them directly to hand.
Space - Excellent, plenty of room for a pram
Price - not too bad for York, just under a fiver for coffee and a piece of cake
Baby changing - clean, but not on the same floor as the cafe
Help - very helpful, no problem providing hot water
Coffee - nice cup of coffee, not the best I've had, by no means the worst
Cake - alas the piece I had was slightly dry, but I wouldn't let that put me off trying more
I'd love to pop in on a night, the bar/restaurant is licensed and there is a nice view of the river.