To be fair, nobody really have a clear picture, or at least agree, on when and how Celtic-speaking people moved in Britain and how the British Islands got Celticized.
It used to be much simpler, as by the middle of the XXth century, it was more or less broadly accepted that mainland populations moved in by the Early Iron Age (ca. 800 BCE to 500 BCE) : after all, migrations seemed to have been the main mode of expansion over Europe from an Alpine *urheimat (*ancestral homeland) : Caesar, providing a source on how Belgians themselves had settled in Britain not too long before his conquests only described a prototypical event having happened elsewhere in pre-Roman Europe.
The interior parts of Britain are inhabited by tribes which by their own >traditions are indigenous to the island, while on the coastal sections are >tribes which had crossed over from the land of the Belgae [in Gaul] seeking >booty. Nearly all these maritime tribes are called by the names of lands from >which they immigrated when they came to Britain. After their arrival, they >remained there and began to till the fields. (De Bello Gallico 5.12)
Alas, archaeological developments questioned too convenient of a model both in British Isles but as well in the whole of Europe to the point the "origin of the Celts" itself remains debated ( u/Libertat).
Britain and Ireland themselves, far from being on the receiving end of Alpine-centered developments, were home to a variety of local cultures that if integrating mainland influences to their own practices (and mostly so on southern and eastern regions) kept on their momentum during most of the Iron Age with even a possible decline of exchange (or at least multi-lateral exchanges) with the mainland during the Hallstattian/LaTenian transition.
Eventually, there is a distinct lack of evidence for a or several massive migrations from the mainland to the British Isles for the Iron Age and, eventually, not since the early Bronze Age when people of steppe ancestry (tentatively identified as speakers of Indo-European languages) moved to Britain and replaced 4/5 of its population in the span of mere hundred years.
It doesn't mean at the latest contact between mainland and the British Isles were unimportant : settlement patterns, prestigious goods, burial practices, etc. and archaeological similarities found their way to Britain trough the Channel and the North Sea during the Bronze and the Iron Ages. How much these implied migrations is up to debate, although it would be safer to neither entirely dismiss them because we can find an explanation that wouldn't include them, neither to give them a disproportionate importance or to understand them as a linear, one-way event. (Fernandez-Götz). More to the point, maybe, would be that these exchanges (either material or humans) might have well created a or many spaces of social and cultural familiarity across the seas.
The Arras Culture of eastern Yorkshire, for instance, is quite remarkable in its apparent complexity : it is distinct in the insular Iron Age by the presence of strong La Tenian influence in chariot-burials and prestige goods as jewellery, and the connection these have with specific regions on the mainland (namely the Marne and Upper Rhine basins) could imply a social if not demographic relations (further stressed by Arras Culture more or less matching the territory of the Parisii, namesake of Gaulish Parisii). Still, as much these populations were materially distinct from their neighbours, they still mixed these influences with indigenous practices or material consumption (either local or traded from southern Britain) with the more diverging material elements hinting at a multi-generational local transformations. Even while archeogenetics elements (notably the presence of R1b-U152) might well hint at migrations impressing on local populations, we don't have evidence for these having a transformative factor onto insular societies, even less so on western and northern Britain.
Eventually, we can somewhat point with more confidence to Belgian migrations into the southern parts of the island, thanks to both Caesar's account but also a set of cultural change brought from mainland (pottery and potter's wheel, urbanization, smith-craft and artistic style, coinage, etc.) more importantly than other mainland influence before, a combination we simply don't have for earlier periods (along presence of cognates as Atrebates, Belgae, Catuvellauni,Novomagius,...). It doesn't tells us much about the demographic and social realities of these migrations (that while debated, are broadly considered as historical), even if it did have a more transformative influence onto "maritime" Britain, notably in the making of the “southern” and “eastern” kingdoms on the eve of Roman conquest.
At best could we say that migrations certainly took place in between 2500 BCE and 100 BCE, but none of them really foundational to the making of "Celtic" Britain, either by themselves or cumulative effects; as most of the material development seems to have been primarily indigenous (at least until the late Iron Age) even while it integrated significant mainland influences and probably human groups into a broad area of geographical familiarity not exclusive and even crossing over others of the same kind, as the Channel or North Sea areas.
