As ever with Roman Britain, the literary sources are sparse and unreliable. The notion of the ancient Britons fighting nude or in light clothing appears to come from various ancient sources on continental groups and to have become a thing that “everybody knows”. Certainly I’m sure I learned it as a child from Horrible Histories, and I strongly suspect I can blame Victorian popular history and art for it, but without diving deeply into classical reception (something I don’t have time for right now) I can’t give a definitive response on that.

For a few continental examples of apparent nudity: Polybius describes a group called the Gaesatae as having discarded their trousers and cloaks for nude fighting, on the grounds that the battle field was full of brambles which would catch on their clothes. He describes this as rather terrifying, but also that their nakedness was (unsurprisingly) a disadvantage. Diodorous Siculus also notes that some Gauls go into battle naked, but does not enlighten us as to who or why. This obviously does not apply to all Gauls, as they are also described as wearing armour of various kinds.

We also have contemporary art produced by the Romans - for example the famous dying Gaul which shows a naked warrior. However, this is a heroic nude with some complex cultural connotations about the construction of the barbarian. It cannot be taken as straight evidence.

All of these examples take place over a wide range both culturally and geographically. If we are to take them at face value (which with any ancient source is not a given; there are always issues of translation and interpretation), then we can say that some Gaulish cultural groups occasionally went into battle naked, but not that this was the general rule for all Gauls or that it was a commonplace occurrence.

There is, finally, one description of the Britons’ clothing in De Bello Gallico, v. 14. This claims that the inland Britons generally dress in skins, dye themselves with woad, and wear long hair and moustaches but shave every other part of their body (!), as well as noting some interesting marriage customs.

Some of this is backed up at least a little by the archaeological evidence; razors seem especially important in British male graves, indicating that they were frequently clean shaven. The idea of them generally dressing in skins seems unlikely; although textile survival is very low in the archaeological record, we can see from the survival of objects like brooches that they were very clearly frequently wearing cloth! It is more likely that this is a Roman stereotype about barbarians he is, for his own reasons, choosing to perpetuate.

Caesar does not, however, note any nudity during his battles against them - which presumably would have been a surprising and notable fact. We would also perhaps expect Tacitus or Cassius Dio to mention it in their slightly later accounts of warfare in Britain, which range over a larger area, and which could also draw directly on primary sources. Cassius Dio especially seems to revel in a sort of tabloid journalism, and “brave Roman legions fight naked Britons” would be a fabulous bit.

Edit: Right after posting this, I remembered Vindolanda tablet 164, which helpfully describes the Britons. It's possible that it reads ne nudi sint (or sunt) Brittones, which could mean literally stark naked, but is more likely to mean unprotected by armour, which is the commonly accepted translation. This is based on other descriptions e.g. Tacitus' Ann 12.35 that the Britons lack armour.

It is likely that the ancient Britons to some extent shared cultural mores with the Gauls on the other side of the channel. Can we therefore say that they likely went into battle nude? As we cannot even say that the Gauls in general did so, or that those who did were nude outside of rare occasions, we cannot. I understand this wasn't quite the question you were asking, and hopefully a Classical Receptionist can come along and tell us how we got here.