Both. I don't think the council euthanises themselves though, I think they're taken either to a cats home, rspca or vet or wherever they go for that. They do rehome them though, I know this for sure because my beau works for the council and he's always wanting the dogs and cats they bring in (usually dogs though I think with a council) because they have their own catchers. They first ask around the office I think and see if anyone wants them, take them through the building, send out emails to be forwarded on etc. And after that they go to the dogs/cat homes.
Just thought I'd ask because I volunteer at the Cat Centre, and I haven't noticed any increase in surrenders. In fact, in the last two days, I'm really the only person who's surrendered any cats. (From a rescue that I did, not my own cats). But I just heard that apparently the RSPCA are now offering cheap/free euthanasia services for cats??! I'll ask at the Centre next week cause I'd really like the council to take the unwanted cats there, almost all of the cats that are surrendered go up for adoption. (Unless they're aggressive or incurably sick.)