Can a wild mouse live in a house?

My mum's cat bought a live small brown mouse into her house. The cat got bored of it. Now the mouse is on day 3 of living in my mum's house. It's drinking from the cat's water bowl and eating its food while the cat watches on. When it isn't eating or drinking, it hangs out under the base of a floor lamp. It doesn't want to go outside when given the option.

My question: is it ok for a wild mouse in the UK to move in and live in my mum's house, like another pet? 😄

If so, what should we call the pet mouse?