"Viable breeding population" just means there's enough cats to reproduce without the whole population collapsing. Frankly, given genetic diversity between individuals, it's likely that at one time the viable breeding population of all cheetahs consisted of maybe two or three individuals. The reason the Florida population of cougars stays as small as it is is in large part because the adult cats keep getting killed, mostly roadkilled, not because of some ecological property keeping them from breeding more and expanding their population. (They also have low fecundity, but this is from over a century of inbreeding, which would not apply to British cats.) Again: I can see cougars surviving quite nicely in the British countryside. But unless they're on birth control pills, I can't see them surviving as legend alone for so long. I realize there are very large areas of England and Wales that are uninhabited; but the thing is, a "very large" area to a human is good enough for a few territories, no more, for cougars. And, again: while cougars include small prey in their diets and can survive on it, their natural instinct is to search for larger prey. In areas with few deer, a cougar behaving normally would then turn to livestock.
While there are dogs that as big as cougars, the tracks are really not at all difficult to distinguish from one another. Dogs have a symmetrical print that usually shows clawmarks, all felids have an asymmetrical print that usually does not. If the people doing the identifications can't distinguish between something as basic as a canid and a felid, I can't imagine they can i.d. a cougar by call or from a distance/blurry photo either. I also agree with the fellow who pointed out that there is nothing in the pictures providing scale, and disagree that a cougar has proportions so radically different from a domestic cat that they can be told without scale. Several breeds of domestic cat (Abyssinian and Traditional Siamese, to name two common ones) have long, lanky bodies, long tails, and small heads- I've got one sprawled on my computer monitor right now, and from a distance, in profile, without scale, he looks cougarish too. (The "lynxlike" cat in the other picture is almost certainly, upon further examination, a Scottish wildcat or a domestic/wildcat hybrid. The proportions, coat, color, and tail are correct.)
Wildcat and wildcat hybrid pics
I've read the reports of the Surrey Puma. Frankly they're so bizarre they fit niether with an actual cougar nor with any other large cat, or in fact any animal that I'm aware of. (3'-4' tall, slitted eyes, big paws, can't jump over short fences and has to scramble over, some mutilated sheep left with a single hole in the side rather than any characteristic cat kill pattern...) The inconsistency and dramatic strangeness of the reports leads me to believe the "Surrey Puma" was mostly created from the imaginations or mischief of the locals.
I still see the escaped exotics/overblown wildcats/active imaginations possibility as far more likely than an established population of any exotic cat species. Other exotics besides the Jungle Cat, Ocelot, and Lynx/Bobcat that are popular as "pets" are the Serval and the Caracal. I don't know about Caracals, but Servals, like Jungle Cats, Bobcats, and Asian Leopard Cats, are close enough to domestic cats to create successful hybrids.