Firstly, these regions were never single, cohesive units. There were no relationships between Gaul and Britain as a whole, because these areas were made up of many independent polities, each with their own political and economic interests.

That being said, there were links between individual polities in Gaul and Britain, through colonization or alliance. Our best source for this is Caesar's Commentaries on the Gallic Wars, especially books IV and V, where Caesar initiates a punitive expedition against the Britons. His reason for this invasion is mentioned in a IV-20:

Caesar...nevertheless resolved to proceed into Britain, because he discovered that in almost all the wars with the Gauls succors had been furnished to our enemy from that country...

What Caesar is saying here is that the Britons had given some sort of help to the Gauls during his campaigns on the continent. There would have been considerable trade between the two regions, and the Veneti in Armorica are said to have been experienced in seafaring, especially along Britain's coast. Elsewhere, he describes how the Belgae who inhabited northern Gaul also established settlements in Britain after arriving as raiders. In 1946, the Irish Celticist T.F. O'Rahilly speculated the the Belgae also colonized Ireland, and are identified in the early Irish text, The Boook of Invasions as the Fir Bolg, because that name allegedly shares the same etymology as Belgae. I believe that O'Rahilly's basic premise was correct; that settlement between the various Celtic speakers of Britain, Ireland and Gaul was common, but his wide-sweeping theory about successive and total colonizations of Ireland by various Celtic peoples relies on too much speculation and generalizations to be taken seriously.

Inter-regional colonization and communication (especially through trade) in pre-historic North-Western Europe seems to have been pretty common in general. This is most evident in Ireland, where you had P-Celtic speakers called the Cruithin (a Q-Celtic corruption of 'Pritani', which is most likely related to the etymology of 'Britain') who were an archaic people inhabiting Ulster, attested in the historical record as late as the early medieval era. In early Irish historical texts, 'Cruithin' also refers to Britons in modern Scotland, which has led some scholars to speculate that the term referred to the Picts, however I believe that the ethnonym referred to Britons living outside of Rome's influence.

Likewise, the Érainn and Laigin appear to have originated as P-Celtic settlers with an identity and language distinct from the Q-Celtic Gaels, who came either from Britain or the Continent. An early medieval glossary of folk-etymologies (and also possibly the first dictionary of a non-classical European language!) called the Sanas Cormaic mentions that the language of the Érainn called Iarmbérla (translated as Iron-Speech) was an extinct and difficult language to understand. I'm still a bit unclear of whether Iarmbérla was a Q-Celtic language related to Irish or P-Celtic language, and it appears that with only two attested words, we may never know.

As a side note, Irish pseudo-historical texts (meaning that they are mostly myth transmitted through oral tradition until being written down in the early medieval period) mention that Úgaine Mór, a 5th century BC king of Ireland, was married to a Gaulish princess named Cessair. While the historicity of Úgaine Mór's reign is questionable, it's not impossible that marriages were arranged between the aristocracy of Gaul, Ireland and probably Britain too. Anyways, there is evidence of considerable contact through trade and colonization between the various Celtic speakers in late-Iron-Age Western Europe. The extent of this colonization is not really known, but we do know that it actually happened.

EDIT: I totally forgot another example from Irish history that suggests colonization by Continental Celtic speakers. Remember when I mentioned that the Cruithin were an 'archaic people'? What that means is that by looking at the names of peoples and polities in early Ireland, we can tell if they were remnants of earlier types of social/political organization. As Irish history crept into the early medieval era, these archaic people with ethnonyms like the 'Dál X' (the gathering of X), the -raige (people of X) or the Corco X (which might mean 'the seed/spawn of X) were displaced and became subjects to the more familiar Ui X dynasties. For the purposes of this answer, the second ethnonym is the most important; Irish historian Gearoid Mac Niocall asserted that these names originated as epithets of whatever pagan deity that those people chose as their principle god, either as a mythical ancestor or patron deity. One of these archaic Irish peoples was named the Bibraige, or People-of-the-Beaver. If you know anything about Irish wildlife, you'll probably think to yourself "wait, but there aren't any beavers native to Ireland!" and you'd be right! This probably indicates that the Bibraige were immigrants from Continental Europe, named after some local deity associated with beavers.