Here is what paleogenetics seems to tell us about ancient Britain:
Neolithic farmers reached the island around 4000 BC and dominated the Mesolithic inhabitants; in 3000 BC the genes of Britain's people were around 80% from migrant farmers, 20% from the Mesolithic inhabitants.
Then around 2500 BC people with the Bell Beaker culture and a lot of Steppes genes reached the island, replacing about 90% of the natives. As we have discussed, they probably brought the Plague with them, which accounts for their population dominance.
New data, published this week, says that in 1400 to 1000 BC a new wave of invaders arrived. This cultural influx is clearly visible in the archaeological record in the form of artifacts and stylistic influence from the continent. These new invaders seem to have made up at least 24% and maybe 50% of the population. As a guess, these may be the speakers of Celtic languages. They also seem to have brought genes for lactose tolerance, which greatly increased in the 1200-800 BC period.
And of course we have data showing that the Anglo-Saxon invaders contributed around 30-40% of the genes in medieval Britain.
The more data we have, the more movement of people we see in the past.
Which means that the old Irish stories recording repeated invasion of the island, and repeated replacements of the population, seem remarkably accurate despite their magical weirdness.