I'm not from the UK, but responding anyway -

When I use the elliptical, I have to wear trainers. When I ride my bike or do floor exercises, no trainers - just socks. It helps me get a better grip on the pedals or carpet. I'm afraid that I'll slide of the elliptical without shoes, though!

I usually work out without trainers, I'll have to try wearing trainers now to see what the difference is!

I've noticed most exercise equipment suggests the wearing of trainers, I'm guessing that's to protect your feet or prevent slippage.

(lol.... look at me talking about "I usually work out".... I should probably say, "on the rare occasions that I work out....")

I think they're actually suggested to protect the ankles - for example specific types of trainers are recommended for running and walking because it's easy to sprain or break ankles. I never wear trainers though, Except when I dance and then I wear high heels because they're good for posture and really work the calf muscles.