According to my Googling the tyres normally in use in the UK are the all-weather type. Winter tyres are just beginning to be talked of by Joe Public, though the remoter areas and the winter sports types have known of them (and sometimes used them) for several years. UK climate conditions are suited to year-round use of an all-weather tyre as a rule, and in most areas.

Apparently there is also something called a summer tyre, but sustained dry weather is so rare in the UK we have no market for those.

I like the above suggested use of LED flashlights, given how long a holdup can possibly last that sounds like a very sensible change from an ordinary torch.

Regarding the retrieval of gripping mats I know that desert drivers using a similar device to escape soft sand sometimes attach the mats to a length of rope and drag them behind after moving off until they reach safer ground. While that would work in principle for snow I cannot see it being acceptable on any road that has other vehicles.

I reckon that one of the most useful things to carry is one or more passengers who can be put to work unsticking you, in the event of getting stuck, and if that doesn't work you can at least share body heat.