The roles of migration, admixture and acculturation in the European transition to farming have been debated for over 100 years. Genome-wide ancient DNA studies indicate predominantly Aegean ancestry for continental Neolithic farmers, but also variable admixture with local Mesolithic hunter-gatherers. Neolithic cultures first appear in Britain circa 4000 BC, a millennium after they appeared in adjacent areas of continental Europe. The pattern and process of this delayed British Neolithic transition remain unclear. We assembled genome-wide data from 6 Mesolithic and 67 Neolithic individuals found in Britain, dating 8500-2500 BC. Our analyses reveal persistent genetic affinities between Mesolithic British and Western European hunter-gatherers. We find overwhelming support for agriculture being introduced to Britain by incoming continental farmers, with small, geographically structured levels of hunter-gatherer ancestry. Unlike other European Neolithic populations, we detect no resurgence of hunter-gatherer ancestry at any time during the Neolithic in Britain. Genetic affinities with Iberian Neolithic individuals indicate that British Neolithic people were mostly descended from Aegean farmers who followed the Mediterranean route of dispersal. We also infer considerable variation in pigmentation levels in Europe by circa 6000 BC.