If all the humans were to suddenly vanish tomorrow (apocalyptic scenario of your choice) which of the UK's common farm animals (cow, sheep, pig, horse, goat) would be able to survive extinction as a species?
First off... i just want to point out that most likely a majority percentage of livestock would not survive, this has nothing to do with there ability to do so, but there ability to get out of enclosures that limit there movement. Farmers often rotate their animals around several fields giving each field time to regrow its grass. with this not being done and the animals unable to get out, the livestock would be fine (ish, it also depends on water access) until they strip a field of its grass and then begin to starve.
Cattle, Pigs, sheep, Poultry, all of these will happily survive in the wild, despite being "domesticated" all of them are still capable of surviving without farmers to tend to them, what will happen however is the traits they have all been selectively bred for will diminish quickly.
Dairy Cows will be uncomfortable for a week or two but would dry up fairly quickly and no longer produce vast quantities of milk.
Meat Cattle or any meat animals would be less likely to have perfect muscle suitable for good quality steaks without a farmer regulating their diet and water content and health closely.
Laying Hens would quickly have a massive boom of new chicks followed quickly by a mass death by starvation, but after that their laying would be reduced as well
Sheep would take a generation or two to slow their wool coat growth rate but that would happen as well. but again one factor that cannot be properly quantified is the shear numbers of dead animals due to starvation laying around which would harbor a lot of nastiness which could cause further deaths to the surviving animals. i don't think there has ever been a study or conclusive evidence as to how that would effect a herd of livestock but its worth noting.
Rare breeds have some of the best adaptations to survive in the variety of terrain and climates within Europe and the British Isles, but they are already in small numbers due to other breeds being better for certain traits that the public wanted, and unfortunately although these are the animals most likely to survive unscathed Human disappearance the best, however due to them being rare breeds, they less likely to survive due to the fact their small numbers mixed with being unable to escape captivity, the most likely Livestock of the rare breeds to survive would be the various breeds of Pig, as any pig farmer will tell you (and i know a few) if you're late feeding them by more than an hour they'll start digging their way out of their enclosures. and often Rare Breeds are not kept in large metal warehouse environments as much as some regular breed pigs are.
Most domesticated animals could survive without humans, at least some subset of the species. The biggest challenge for them would be getting "free" of artificial enclosures that humans have put them in.
Those animals that would do best are sheep, goats, pigs, and chickens. The first three because their species have been historically allowed to range free, unlike horses and cows, and the latter because chickens have a very fast reproductive and maturity cycle. Cows tend to be handled on a more industrial scale, and are more likely to subsist on feed than sheeps, goats and pigs. They may have become too far domesticated to easily survive as a species on their own, but I would imagine even they would survive in some numbers. Horses are a real concern. Since they are a pleasure animal, not a food animal, they tend to be housed and bred in smaller numbers. This means there are fewer of them overall and in less dense concentrations. It would be far harder for them to survive as a species after humans vanish because of this reason, but a breeding farm may lead them to surviving.
Another consideration is the lack of viable males among the first generation. Many male farm animals are slaughtered young or gelded, leaving one or two viable males. This causes a vulnerability point to the population, because if the one viable male dies, there's no males to fertilize the females of the herd, and the herd dies after a generation. If the herd survives to the second generation, more males will be born and the issue will resolve itself naturally. The popularity of artificial insemination among animals aggravates this problem, as some small farms may not even have a viable male, relying on imported semen from another farm.
A final interesting point is the genetic one. Most domesticated animals have genetic traits that are not suited for the wild. Black Angus cows, for example, have very small horns. They are basically useless in the wild. Most food animals have also been bred to grow fat quickly and be slaughtered, so who knows what kind of long-term genetic diseases have crept into their genetics, or issues they will have if they switch their diets to a fully natural one.
Some pigs will certainly survive. Feral pigs are know worlwide, and most pigs raised in Europe are not that far from their wild ancestors. Early navigators used to leave pigs on newly discovered islands, and they always thrived.
Goats and some sheep will most certainly survive. Feral goats are know worldwide, and there are islands in the UK where local varieties of sheep are left pretty much to their own devices. Early navigators used to leave goats on newly discovered islands, and they always thrived.
Some cattle will survive. Unless the UK is a very special place, there must be primitive varieties of cattle, who are perfectly able to take care of themselves. There are actually some private parks in the UK where they keep feral cattle; so we already know that cattle can live on their own on the island.
Some chickens will probably survive. Domestic chicken are barely different from their wild ancestor, and feral chicken are known worldwide. They are basically tropical birds, though; so their numbers may be limited.
Ducks and some geese are very close to their wild ancestors; they will do just fine.
Some dogs will survive, especially the larger and less stupid races. In actual reality controlling feral dogs is a problem.
Cats will thrive. Cats are anyway wild animals which for their own reasons choose to tolerate humans; they do very well by themselves worldwide. In actual reality controlling feral cats is an unsolvable problem.
I don't know about horses. In the absence of humans the UK lowlands are a vast forest, not ideal wild horse country.
The only livestock that I think won't survive without humans is sheep. Why sheep? because sheep have been bred to grow a huge amount of wool and if there are no humans to remove the wool they just become giant grey clouds of wool. This impedes movement, overburdens the animal and can cause it to overheat.
The only other animal that might have problems would be a dairy cow because they produce so much milk that humans have to milk it for it to be comfortable. Atleast that's my understanding of it.
The rest should be fine because I don't believe they have been selectively bred as much as sheep and cows were for certain traits. Please remember these are farm animals, not wild variants that I am referring to.