I have heard both first hand and second hand accounts about wild big cats roaming the UK. Some have come from people that would be considered reliable witnesses, and I include park rangers and zoo people in that category. But too many times it is shown that people have mistaken domestic cats or foxes for 'black panthers' (foxes mistaken for thylacines in Australia are another example). Commonly the witnesses have misjudged scale because of poor light or some other reason. There are so few examples of true feral wild cats, excluding Felis catus. Yes, there are a couple of isolated individual cases of lynx and jungle cats found in the UK which have been proven, or surmised to be, escaped pets. And they can appear to survive quite well in UK conditions. Whatever the case for their freedom, they were never part of a breeding feral population, nor were they large predators like leopards or pumas.
What I also find intriguing is the longevity of these rumours. Big cats can live up to around 20 years in captivity, but 12 years would be more normal in the wild. But the stories of some of these cats have persisted for multiple decades, indicating not one lonely cat but multiple individuals which are able to meet up, procreate and live full lives under the noses of people.
While I agree that big cats are notoriously difficult to see in the wild even within their normal range when living close to humans, it is certainly not impossible to have definitive proof that they are existing there. I once heard an interesting talk about tiger researchers in dense rainforest in Sumatra who had built up a good knowledge of all the individuals within their study area, but had never physically seen one of their animals! From my understanding camera traps, roadkill, pug marks, scat and fur samples have all failed to show definitive proof of wild big cats in the UK. Happy to be proven wrong.