Ancient historians are sort of moving away from defining Roman era religions as 'entities' that can be either Roman or British. Pagans of the time could adopt new gods as they came in touch with new cultures but it did NOT mean that they abandoned the gods they had worshipped since their childhood - imperial Romans themselves sacrificed to non-Roman gods all the time, and this did not do anything to reduce their 'faith' to their 'own' gods. The modern word for 'religion' applies very poorly to the polytheistic pagan religions of the past.

Our knowledge of pre-Roman British religion is almost non-existent: there was no writing systems before Romans, so we just have to make guesses about what British people believed in based on enigmatic grave finds, monuments, and pieces of art. As soon as Romans came to Britain, Roman style altars and temples to Roman gods appear in Britain. After some time (very soon after the Romans came, I'd presume; dating inscriptions is a pain and sometimes impossible, especially with altars as the usual formula did not include a date), elsewhere unknown names of deities start to appear on these Roman altars which probably have British origins. Here's a list of about seventy such deities (the website otherwise has somewhat out-dated ideas about Roman religion!). Veteris, Coventina, Cocidius and (Minerva) Sulis are the best known Romano-British deities.

Of course, there's a number of problems of using these altars or temples to 'indigenous' gods to deduce just how 'British' the religious landscape of Britain was. The fact that the names of these gods are found carved on altars means that the gods are already given a Roman form of worship: the act of writing is Roman, the language is Roman, the formula is Roman, the shape and the very idea of a monumental stone altar to a deity is Roman. Secondly, very rarely can we say for sure that the dedicator of the altar was a native; Romanised individuals often adopted Latin names, and, like I said, Romans themselves venerated local gods, too. Also, names with Celtic roots might just as well come from Europe, and a LOT of soldiers came to Britain from Gaul, Belgae, and Germania. Thirdly, altar's were a luxury of just very small part of the society and gives us only a tiny glimpse of the religious lives of the recorded individuals, too: what about all their other rituals and beliefs that do not leave tangible traces such as stone altars or temples? And, fourthly, we rarely can say for certain that any 'British' god was worshipped in Britain before the Romans appeared. Take Minerva Sulis, who had a sanctuary and famous baths in the modern Bath. 'Sulis' is definitely a word of Celtic origin, but, she had a Roman style temple and she was worshipped for her purifying baths when public baths are obviously a Roman import, and she received a lot of curse-tablets which also come from the Greco-Roman tradition. Is there anything to suggest any continuity from the worship of some British deity called 'Sulis'? Not really, but it's entirely possible there used to be one, and because of her similarity with the Roman Minerva, she started to be worshipped as 'Minerva Sulis'.

So, the Roman influence on the British religious landscape was pretty instant, but there are regional variances. Some probably only kept worshipping their own gods, majority perhaps worshipped a mixture of 'imported' and 'indigenous' gods, and probably through methods and rituals that were equally hybridised. There's for example some archaeological proof from Gaul of continued human sacrifice rituals (which the Romans never did and strictly banned) at Roman altars for a Roman deity, Hercules.