Depends on which Brits you mean. The Scots and Welsh (and Irish) are nearer akin to Basques than anyone else. But the English (Anglo-Saxon) newcomers, are indeed close to the Dutch.
The Britons described in this essay, are the ancestors today's Welsh. They did indeed adopt quite a bit of Latin language and culture during the Roman occupation, but genetically they apparently didn't change much during that time.
This includes my own roots in Galicia. Galicia is at the farthest northwest corner of Spain and was a Celtic region during Roman times. Thus it's Roman name "Gallaecia" meaning "Land of the Celts".
The genetic links between Galicians, Basques, Scots and Welsh suggest they were all part of the same initial edge of a demographic expansion towards Europe.