I can corroborate. I grew up somewhere where we heard cougars regularly (for those of you who don't know, they scream at night - it's a distinctive, spooky sound), and spent a lot of time in the woods. My ambition was to spot one, but though I found scat and paw prints and deer I was pretty sure they'd killed I never saw one in the flesh.

A neighbor found one dead, one time. (Dead from illness; I never heard of one being hit by a car - I think they're too smart for that.) A few other people I know were luckier, and had sightings, all of them too brief to photograph.

I'm skeptical about British big cats, but think it's plausible that (if they exist) they could remain all-but-invisible, and agree that corroboration would be likeliest by spoor or DNA, not by sight.

Edit to add: My totally amateur guess is that they wouldn't be cougars, because they scream, and I've never seen anyone reporting hearing that in the UK; and they're not likely to be leopards, because they hang out in trees, and are relatively easy to spot when they do. I'd say jaguars, which do neither, and occasionally come in a black variant. I'm still skeptical, but that it is remotely plausible - magnitudes more plausible than Bigfoot! - makes it an intriguing story.