I think a more accurate question is when did they converge - while they share a landmass, historically the island held a number of linguistically-distinct ethnic groups and smaller kingdoms.

I agree with catnik that the proper verb is convergence. The 3 peoples you list are descended from 3 different groups of invaders.

First inhabitants that we can trace are the Britannic speakers, which divide into the ancestors of the people in Ireland (originally called Scotia) and possibly the Picts in Alba. They arrived in the 3rd millenium BC. The Brittonic speakers, ancestors of the Welsh, Cornish &c, developed in the south of Prittania. These were the Britons that the Romans invaded and ruled.

In the 5th C, the Angles, Saxons & Jutes are recorded to have begun invading Britain, eventually driving the Romano-Celtic culture into ths west, like Northumbria, Wales, Cornwall &c. These Anglo-Saxons are the ancestors of the English speakers. The very name "Welsh" is AS for "foreigner, outsider, not one of us."

In the meantime, the Scots of Dal Riata/Dalriadans were sailing out of what is now Ulster to take over the land facing west in Alba. Eventually they merged with Pictish kingdoms before the 8th C to form the Kingdom of Alba.

And so the Welsh, English, & Scottish have different identities because of their different historical & cultural origins, not because of divergence from a common root.