Time to stop feeding the cats Rupert!
I live in the opposite building from Rupert and we suddenly started to have mice about 4 years ago after 15 years with no issue. Don't know if that was caused by a neighbour who used to keep a couple of birds and had lots of bird food lying around they could feed on.
Anyway, I despaired. They don't touch poison or traps, they are very clever creatures. I eventually got sticky pads and caught a couples of them (yes it is awful) but then I actually saw them jumping over the pads. They learn from their mates' fates (even when I lifted some boards and put some traps there, they did not go for them, they must have infrared eyes).
In the end, I blocked all holes even as tiny as a pen tip. Also I put all food away in plastic containers. They did not appear anymore but I could still hear them under the floor boards so I got some of these noise emitting plugs from Amazon. It seems to work but I can still hear them sporadically at night and it drives me crazy...
Last year, I had a room decorated and whilst the carpet was removed, I put some wire wool in between the floor boards (and dropped heaps of poisons through the gaps, my dad told me to use gloves as mice can smell human handling of poisons). One day I noticed that they had removed some of the wool and one mice got away (I caught it later with a glue pad and a maltezer). Once installed, I think they run from one property to another under the floor boards and prosper...
Another idea, check if you have airing bricks in your walls that could be letting them in.
Unfortunately you can get rats and mice in the same property, the people on the ground floor in Styles had both when we had a rat infestation. The rats seemed happy to coexist with the mice but perhaps we have particularly good community relations.
They came in, baited to eliminate the immediate problem and then did a really thorough examination of the house for any potential entry points and blocked them all. So far we've had two years (and two winters) without any mice, so I definitely recommend them!
No connection other than being a satisfied customer.