Gaulish or Galatian may have survived up until about the 6th century AD - there are anecdotes and inscriptions from that period. The predominant or official language of the Belgae in Europe was almost certainly celtic and very similar if not almost identical to galatian, although large parts of the region occupied by the Belgae close to the rhine was supposedly a melting pot of Celtic and Germanic cultures.
It took a long time for Latin to completely usurp galatian, which was viewed as a barabaric language by the Latin speaking elite. The reason that the Brythonic languages survived is simply because the Romans were not here long enough and perhaps with some exceptions of the iceni and the silurians Britain pretty much accepted their fate. Britain was invaded by the Romans after the peak of the Roman empire and most of Britain after witnessing the relatively simple defeat and destruction of Gaul and the Belgae by Julius Caesar would have been mindful that resistance to the Romans would have been met with no mercy, whereas the Romans would be benevolent to those who accepted what was coming.
In my mind you can imagine parallel British and Roman societies in Roman times - acceptance of the Romans, who had controlled trade in the Mediterranean for centuries and now controlled the rest of Northern Europe, meant that trade was only possible with the will of the Roman Empire. The Romans were very hierarchical and it served their purpose for the elite and easiest to control to be Latin speaking and I can't see how or why they would have been overly concerned about the language that the plebs used.
It also suited the Romans that Britain was dependant on Rome for military support - this is the way they had grown their empire in the first place, so that when the Romans left, they left a country that was very weak militarily and open for attack. It was almost inevitable that Britain would be invaded after the withdrawal of the Romans, but according to Gildas the first attempts by invasion from the Saxons was repelled and they didn't return for another century - something corroborated by the writings of Bede.
It is unlikely that the invaders to Britain annihilated all and everything that they conquered - it is far more likely that plagues and diseases over the following millenia killed far more people than any conflicts. As for genetics that is totally misleading - we are all Europeans and although the 19th and 20th century was full of people trying to show distinct European races, that is simply bonkers. We have had very diverse cultures and isolation/migration paths have lead to some genetic differences that give some genes more dominance in some parts of Europe than in others, but nothing to be regarded as of any real significance. In my mind culture, traditions and language are far more influential on our societies than our genetic make-up.
The use of the words Keltae was used by the gauls to describe their language - at least according to Julius Caesar - someone who in his words seemed to have singly handedly conquered Europe killing thousands of people with his bare hands.
Julius Caesar's legacy was the ending of Celtic domination of civilisation and culture in North Western Europe, but it wasn't complete and lived on in Armorica, Western Britain, Scotland and Ireland as it does today.
The celtic languages alomst certainly go back to the bronze age, when it would have been an indo-european dialect. Luckily the legacy stills lives on in the Brythonic and Goedelic languages and celtic traditions - albeit that many of those were reinvented by some romantic individuals in the Victorian era.