My turntable is a linear tracking model, where the tone arm moves like that! But it doesn't have a laser stylus, just the plain old physical one. Which probably needs to be replaced since I bought the unit at a thrift store and I have no idea how much use it's had. Came with the original manual though! It cost me about $10 or $20.

It's a Realistic Lab-2000, which is a store brand for a company called Radio Shack. They did have some stores in the UK, but I don't know how well known they were there.

https://www.vinylengine.com/library/realistic/lab-2000.shtml