A real giveaway is the frequency of sightings around 'less respectable' collections. In the UK, lynx have often been spotted around certain parts of Norfolk, not far from the (now closed) great witchingham wildlife park (which would regularly advertise surplus lynx in a cage bird hobbyist's magazine), or on Dartmoor (where, for years, the local wildlife park under its previous owner deliberately bred large numbers of jaguar, tiger and puma to attract visitors). Either escapees stay in the area and aren't reported, or surplus animals are released, or possibly feral animals are attracted to the area to mate with the captive stock. The new owners of Dartmoor wildlife park claimed on the recent documentary series that their remaining two female pumas would often start calling at night when in season, supposedly to feral males in the area.
Many of the species caught in the UK were just escapees, and there was never any chance of a population being established. The clouded leopard was a known escapee from howletts, the servals on the Isle of Wight were likely escapees and did not survive long. Interstingly, Jungle cats have often turned up as roadkill around the southern ports, I recently read that sailors for the old east india company would take this species from Indian villages on board to control pests, and then presumably released them once they returned to England.
Given the behaviour of the press in relation to this issue, the longer the authorities can continue to dismiss or cover up evidence, the less chance of a public outcry forcing them to 'do something', compensate farmers etc. If it isn't real, it doesn't cost, basically.
If released/escaped cats are to be successful, they have to hunt, obviously they are going to do so at night. Searching for a black leopard or a puma in a rural area at night is like finding a needle in a haystack. All they have to do other than this is lie low during the day. It wouldn't be hard to remain undetected. So you are left with anecdotal evidence, occasional tracks/droppings and cat-like livestock kills.
And we are a tiny island in comparision to you