grey squirrel, being relatively recent, is probably the best known of these, although the brown (or "sewer") rat is another, only arriving in the 16th century.
Another example was first introduced in Roman times, and so has been here rather longer - but, still, it could be argued that it's not technically native. This is the rabbit (Oryctolagus cuniculus).
The word "rabbit" is itself interesting. Although French is another exception, the word for "rabbit" in most European languages is some variant of the Latin name for the animal, "cuniculus". This used to be true in English too, with the word "coney" being the standard term up until the late Middle Ages. The problem was, that was supposed to rhyme with "money" which meant that, by around the Rennaisance it sounded exactly the same as a certain rude word of the time (derived from another word that remains highly offensive today). So, to avoid sounding like they were swearing, people switched to the slang terms "rabbit" and "bunny" instead, and the first of those eventually became standard. Even today, with the swearword largely having faded from use, on the rare occasions when we say "coney" we rhyme it with "pony", and often cut the 'e' out of the spelling to make that clearer.
In fact, the native habitat of rabbits is warm, dry grasslands mixed in with patches of bushes or scrub. Which would be a problem if you ever introduced them to, to pick an entirely random example, Australia, but isn't really a good description of the British weather or the heavily forested pre-industrial landscape of the country. While they may once have lived further afield, during the Ice Ages, they died out everywhere except southern Spain and Portugal, and probably northern Morocco. By the time the Romans came along, they had spread north to western France, and east into Algeria, but that was about it.
The Romans, however, probably discovering these animals after their conquest of Spain, found them to be tasty, and transported them widely across Europe. Where, as it turned out, they were willing to put up with cold and rain rather better than one might expect. Today, rabbits are considered native across pretty much the whole of western and central Europe, as far north as the Shetland Islands and as far east as Moldova. Scandinavia has mostly proved a step too far, but they are found in Denmark and southern Sweden, and while they are rare across most of the Balkan mainland, they are widespread on the Mediterranean islands, as well as Madeira and the Azores.
Some of these island populations are formally regarded as distinct subspecies, but it's unclear how many of the six recognised subspecies are really distinct from one another. On the other hand, there is some evidence that the rabbits of North Africa are already some way towards forming an entirely separate species from their European kin.
That, of course, ignores the presence of modern domesticated rabbits essentially worldwide, with the British Rabbit Council currently recognising 95 different breeds. By most estimates, they are the third most common domesticated pet worldwide, following cats and dogs. Inevitably, some have escaped and gone feral in places outside their natural range, while wild rabbits have been introduced to every continent except Antarctica, with varying levels of success.
Despite this, and the fact that rabbits are an invasive pest in many parts of the world, back in their native homeland they are becoming increasingly rare, with the Spanish population having fallen by a massive 80% in the last quarter of the 20th century. As a result, rabbits are officially regarded as "Near Threatened" species, even as they continue to expand across many areas where we don't want them. While habitat loss is a factor in this decline, disease has been far more important, with myxomatosis and, more recently, rabbit haemorrhagic fever (which is related to Norovirus in humans) having killed off substantial numbers over the last few decades.
Rabbits feed primarily on grass, although they also eat various herbs, with the exact proportions varying throughout the year in order to maintain a high-quality diet as the seasons change. Unsurprisingly, this also varies with location; rabbits living on the Norfolk coast prefer to eat red fescue, and avoid marram grass, while those living in similar habitats in Portugal actually like marram grass... presumably, it all depends on what the alternatives happen to be.
The real reason that rabbits prefer dry, but not overly arid, climates is that these provide the best soils for burrowing in. Rabbits are naturally sociable animals, with multiple females sharing a single warren wherever possible, although, as one might expect, lone females are more common where the food supply is limited. This means that warrens can be large, although the details of their layout may depend in part on how suitable the soil is for digging, with those in sandy soils being simpler than those elsewhere, yet still roughly the same overall size.
Since one of the primary purposes of the warren is to raise young, it's the females that spend the most effort in defending them. The females are not necessarily related, and typically form about 60% of the adult population in any given breeding group - the other males are solitary and may not even have small resting burrows, spending the day hiding in heavy vegetation. The females also have a strict system of social dominance, with the higher status individuals having more young than their compatriots.
Rabbits are well-known for breeding like... well, rabbits. The breeding season for rabbits is essentially "whenever there's enough food to raise a litter without starving", and can last anything from four to nine months of the year, depending on the local climate. Males seem capable of breeding year-round, and pregnant females do have the ability to resorb some or all of their embryos if the weather changes for the worse or they are simply about the give birth to an excessively large litter. Come to that, some females will kill the young of their fellows if resources for the collective group are in short supply.
Pregnancy lasts for about a month, and the young are weaned at three weeks of age. Since females can become pregnant again a week or so after giving birth, this makes it possible to have several litters a year. With an average litter size of around five young, this means that a single rabbit can give birth to anything from 15 to 45 children each year. Under normal conditions, the mortality of young rabbits is quite staggeringly high, but if there are no natural predators and plenty of free territory, rabbits really do multiply very rapidly.
When I have said "rabbit" so far in this post, I have been referring only to the animal originally known by that name, and still widely called that in Europe. More accurately, however, it is the "European rabbit", since many other species exist, mostly in North America. The closest relative of the European species is probably the so-called hispid hare (actually a rabbit, not a hare) which inhabits patches of tall elephant grass in the foothills of the Himalayas.
The larger group to which these animals both belong probably originated in North America during the Middle Miocene, around 15 million years ago. Four million years later, some of these early rabbits crossed over to Asia, from whence they gave rise both to the European rabbit and hispid hare, but also the riverine rabbit of South Africa and the odd-looking short-eared Amami rabbit which lives only on a couple of small isolated islands south of Japan.
The ones that stayed behind in America, however, gave rise to many different species, most of which are referred to by the collective name of "cottontails" (Sylvilagus spp.) These are found across virtually the whole of the US and Mexico, with a smaller number of species in Central and South America. (There are a couple of northern species that cross the border into Canada, but not by very much). Compared with European rabbits, cottontails are typically antisocial and they don't dig burrows, let alone warrens, making do with shallow scraped-out depressions in the soil.
Even including all the cottontails, however, this group accounts for less than half of all the species of the rabbit family. And, if rabbits are not, historically speaking, native to Britain, it turns out that two of the other species are. These, of course, are the hares, and it is to them that I will turn next.
[Photo by "Aiwok", from Wikimedia Commons. Cladogram adapted from Matthee et al. 2004.]