The Brittonic speakers later became the Welsh, Cornish, and (as emigres to the continent) the Breton.
Very different branches of the Celtic language tree. Not mutually intelligible. Brittonic has more in common with (extinct) Gaulish ("P celtic")
Also the Romans established a landholding system in Gaul that took its agricultural lands under Roman control, so in a sense they definitely had impact on the local population. The rich farmland of Gaul was one of the key reasons for the conquest in the first place.
Though service in the Roman military by Gauls themselves meant they could in turn have rights to access the land as Romans. A further method of Romanization though.
I've lived in Dublin, the Isle of Man, and Cardiff - really interesting hearing the language and the differences. I even tried learning Manx Gaelic, but didn't get very far.
Both also seem to have crazy orthography, too, hard for an English speaker to figure out
Manx and Irish Gaelic is just-about mutually understandable, though with difficulty. It's like a Londoner trying to understand a Glaswegian.
I gave up trying to learn Gaelic after learning that there are no words for "yes" or "no". To agree with someone you have to construct a positive sentence agreeing with them (e.g. "sure and so it is"). This suddenly made so many Irish-English colloquialisms make sense :)
The language changed over decades as the upper classes became Romanised (and in the same way Normanised a thousand years later), with Latin/French becoming the courtly language and Gaelic/Saxon the commoner's language. Until finally English emerges as a bastardised combination of all of them.
Whereas in the Danelaw the invaders changed the language and culture in a single generation wherever they took over. Though it's interesting that on the Isle of Man the Viking raiders only killed the men, and settled down on the island with the Manx women, resulting in a Gaelic-speaking Danish culture. There are places in Scotland where this happened too. It's a different combination from the Danelaw, where the settlers brought their own women and the English were displaced completely.
"Britain" is a Romanised corruption of the pre-Roman Gaelic name for the island, "Prydwen". The British identity was already well established before the Romans. They became the "Romano-British" under Roman rule, still speaking Gaelic, and then the Welsh as they were displaced by the incoming Saxons (and still speaking Gaelic!). The Welsh identity is definitely a product of the Saxon invasion, as you say, including the massive still-smouldering resentment of the Saxons (they refer to England as "Lloegyr" which I was told by a Welsh historian means "lost lands" but that doesn't seem to be accurate).