That's a very interesting question, what language was spoken in Doggerland and presumably surrounding border areas.
What we know is that some sort of 'celtic' language is spoken on the far western margins, Ireland, Scotland, Man, Wales, Cornwall, formerly Cumbria and lowland Scotland, Brittany etc. I don't know whether they've resolved the question of Pictish yet.
We also know that Germanic languages were and are spoken in eastern Britain, England and Scotland and western Europe, Holland, Belgium, Denmark and Norway. Dutch was also spoken in northern France and north western France.
Which makes Doggerland possibly a Germanic speaking area.
We know that western Britain and Ireland were populated by tradition with people from Iberia and from for example Troy and Egypt. This is a very strongly held tradition which is of course ridiculed by academics. But we do know that there was a great deal of trade between all these areas and the Mediterranean in the copper, tin and other metals trade. So the two areas were known to each other on a commercial basis.
New genetic findings tell us that the main dna in Britain and especially in the west is the R1b type which goes through the Mediterranean into the area of Georgia. A relation in R1a took a different route by heading through eastern Europe to northern Europe and Scandinavia.
Linguistically the celtic languages share similarities with the hamitic group of languages especially with verbal order. Verb comes first followed by noun or pronoun.http://www.academia.edu/283231/Remarks_ ... c_question
Here is an academic paper which goes into this in some detail. Although it must be said that most academics ridicule the idea.
If the original settlers of western Britain left Spain by ship, which is what Irish tradition tells us the natural swing westwards of the French coast would tend to force them to the extreme west because at the time of Doggerland there was a lot more coastline heading in that direction. But those heading in a more direct northerly migration, following herds perhaps would quite naturally move straight into Doggerland and eastern England.
There is also the possibility that there were languages already being spoken in the area which had no relation to the celtic or Germanic languages. And also that there were a mixture of languages which eventually morphed into the more recognisable languages we know today.
What we do know is that from the celtic languages across to Flemish and Dutch these languages are very gutteral, with the exception of English which may point to a more Norse origin or connection. Perhaps living in a perpetually damp climate at near sea level affects the vocal chords?