From both a narrative (general history) and scientific (genetics), the answer is No (there was not a lot of interbreeding).
(We get more precise as we go from narrative history to genetics, as shown below -- but science requires certainty which creates some confusion in narrative history).
The narrative history of Roman Britain, was fairly straightforward until the pesky thing called science (genetics, archaeology, etc) started the confusion (with males vs females, slaves vs nobles, true romans vs barbarian romans ... i.e. what do we mean by romans?, which part of Britain?).
Before the science, in terms of (classical) narrative, I believe the short answer is No, there should not be much intermarriage (legal interbreeding). This was the general belief because of several factors:
- Roman Identity - this was a prized possession and not everyone could become a citizen (via marriage)
- Rights of Women - was less much than the modern world (i.e. right to vote, etc.) but one crucial aspect of having a mother who was a (barbarian) citizen was her children would also become citizens
- Rights of Inheritance - property (land and chattel) were not taken lightly and even wills of Roman soldiers had to get approval.
In short, the social structure of Roman society was not to inter-mix with the barbrians of the provinces. Here is a speech by Claudius that did not go down well at the senate because he wanted to admit noble citizens of Gaul to the Senate:
Once upon a time Kings ruled this City, but they were not fated to have
home-grown successors. Outsiders took over their rule, foreigners in fact,
for when Numa succeeded Romulus he came from the Sabine lands—not
far away to be sure, but it made him a foreigner in those days. When
Tarquin the Elder succeeded Ancus Marcius, well he was of mixed race, for
his father was Demaratus the Corinthian, while his mother was born in
Etruscan Tarquinii. She was not a wealthy women, as you might imagine
given she had agreed to such an inferior marriage, and for that reason he
was unable to hold offi ce at home. But he migrated to Rome, and here was
made king.
and (the factors as explained above)(emphasis mine)
During their (peregrini) twenty to twenty-five years of service, soldiers of all kinds naturally formed relationships with local women: but these were not formally marriages and children born to them would not be citizens. Special dispensations allowed soldiers to make wills, but their wives and children had no automatic rights in respect of them. An auxiliary veteran could make his slave a citizen by freeing him, but any children he had fathered before he was discharged would have to join up themselves if they wanted the same status.
Source: Rome: An Empire's Story (Oxford, 2013), p.218 and p. 222, respectively.
(On sources, as per your request in comments) The main sources of (classical) history of Roman Britain should include R. G. Collingwood from 1923, and less old but still excellent, both 1981, Roman Britain by Peter Salway (Oxford) and Roman Britain by Malcolm Tood (Fontana); both books excellently reviewed together by the LRB (worth a read). The latter was also editor of A Companion to Roman Britain (Blackwell, 2004).
The answer from a scientific perspective is a somewhat more involved, and difficult to make a simple and clear-cut statement (as indicated by your articles) because we have to differentiate between male and female lines. But in general, the conclusion is No, as there is very little evidence of interbreeding for both maternal and paternal lines.
According to the world's first genetic archaeologist, Bryan Sykes, a Professor of Human Genetics at the University of Oxford, who conducted extensive DNA research within the British Isles, covering a much longer (earlier) history from pre-history to modern-day (emphasis mine):
The first conclusion, blindingly obvious now I can see it, is that we have in front of us two completely different histories. The maternal and paternal origins of the Isles are different. And that should be no surprise, given the opposing characters of the chroniclers. The matrilineal history of the Isles is both ancient and continuous. I see no reason at all from the results why many of our maternal lineages should not go right back through the millennia to the very first Palaeolithic and Mesolithic settlers who reached our islands around 10,000 years ago. ...
and (for females)
Lastly, I have found a tiny number of very unusual clans in the southern part of England. Two of these are from sub-Saharan Africa, three from Syria or Jordan. These exotic sequences are found only in England, with one exception, and among people with no knowledge of, or family connections with, those distant parts of the world. I think they might be the descendants of Roman slaves, whose lines have kept going through unbroken generations of women. If this was the genetic legacy of the Romans, they have left only the slightest traces on the female side...
and (for males)
I have tried to find Roman Y-chromosomes, but they left very few traces that I can be sure were theirs. Only one very rare patrilineal clan, without even a name, may be the faint echo of the first legions. It is found in southern Europe, including Italy. What makes me think, as well as this link to Italy, that it might be linked to the Romans is that it is entirely restricted to England. There are no traces beyond the borders with Wales or Scotland. There may be others, but as was pointed out, the tradition of recruiting legionaries and auxiliaries from Gaul and other parts of the Empire, as well as from Britannia itself, makes them very difficult to spot among the descendants of later arrivals from the same areas. But true Roman genes are very rare in the Isles.
Source: Blood of the Isles (Penguin, 2006), Chapter 18 (final chapter).
