Did they just blend in with the Romans and Anlgo-Saxons?
They blended into the genetic stock. Just as all Romans the "Romans" of Britain largely traced their genetic heritage locally, they weren't imports from across the seas.
In terms of migrations of different peoples to Britain, you have several different waves: stone age peoples, neolithic longhouse peoples, beaker peoples, celts, romans, britons, angles, saxons, irish, norse, normans, etc.
According to genetic studies, the predominant sources of ancestry for the average Brit came to the British Isles during the stone age (specifically a period between around 15000 and 7500 years ago, during the mesolithic/neolithic). If you trace back the ancestors of every person living in Great Britain currently you would eventually end up tracing about 3/4 of those lineages back to folks who traveled to Britain by foot, over land bridges that existed during the last great ice age, originally from the Basque region of Spain/France near where the Pyrenees meet the Atlantic Ocean.
Source:
I just wrote a long response to this, then accidently pulled the plug out from my computer, so this will unfortunately be a shorter version.
A notoriously complicated matter, bedeviled by politics and wishful thinking. This question can best be answered by genetic study of the modern English, and comparing them to the modern Welsh. There's is no debate at all that the welsh are the descendents of ancient Britons (as are, to a lesser extent, people from Cornwall and also Brittany, in western France). But interpreting genetics is itself a bit of a minefiield, as differnt scientists and geneticists interpret different genetic markers in divergent manner. It's still a bit of a new science.
To answer the first part of your querry, there is very little evidence of a Roman input into the genetic background of Britain, although there may be some...http://www.jogg.info/32/bird.htm
The big question is about the Anglo-Saxon invasions. Broadly speaking, there are a couple of different models. One postulates the near total ethnic cleansing of the original inhabitants of the southern British lowlands by English invaders. So they invaded, killed off the vast majority of locals, kept a few as slaves, then slowly settled the land as they expanded in population. The other says that the invaders were very few in number, and they just moved in as lords, ruling over majority British population, a bit like the later Normans. According to this, native Britons adopted the ways and lifestyles of their new masters, and slowly became the modern English.
There are different answers for distinct regions. Roughly, the further west you go, the more Britons remained. So in a places like Essex, far to the east, there were probably very, very few left. In Wessex, south western England, there are laws and charters describing the existence of a series of classes of Welshmen, which indicates there were lots around, although they had fewer rights than Englishmen. West of the Pennine mountains, in the north, the population was majority British/Welsh until the Viking invasions.
A few things are not in debate. Firstly, there was certainly a major decline in population levels from around 300 ad, which accelerated with the withdrawal of the Romans. The plague of Justinian hit hard in the 540s. Secondly, there are distinct changes in the archeological records which indicate people were living differently. Thirdly there was mass emigration from Britain to Brittany.
The number of actual English/Anglo-Saxon invaders probably wasn't all that high, and they came in waves over the fifth and sixth centuries. Even the highest estimates don't go much further than a few hundred thousand.
However, the thing to bear in mind is that the area itself really isn't very big. In the north east a few settlers likely ruled over a large population of natives. In the west and far west, there were still may Britons around. In the east there were very few. The western midlands remained British until around 650.
So the big question revolves about what's left, the south and the east/central midlands. This is geographically a fairly small place, with no mountains to hide in, no strategic passes where a small force can repel a much larger one,and criss-crossed by good Roman roads. So ethnically cleansing that area, driving out and killing all the locals over the course of a half century or so, isn't all that unimaginable.
There's some genetic evidence that this is what happened...http://mbe.oxfordjournals.org/content/19/7/1008.long
This article compares populations in a few English towns and compares them to 2 towns in Wales, and finds definite genetic differences. It's not exactly easy reading, and that's the thing with genetics. It doesn't simplify itself unfortunately.
There's a lot of work that still needs to be done on this whole debate, but things are gradually becoming clearer as more studies are undertaken. It's a difficult subject, particularly as the archeology of Romano-Britain is very difficult to differentiate from later and earlier periods, and genetically it's almost impossible to tell the difference between anglo saxons and later viking invaders. (It also means historians have to grapple with a successful British counter-offensive against the invaders in the early sixth century, which led to a 50 year odd British ascendancy The trouble is, for historians who worry about their reputations, that means dealing with a figure who later became known as King Arthur. Naturally, they don't want to go there.)
It was a partial blend. The time at which the most diverse peoples would have reached the landmass would have been around the point of Roman occupation. The Romans had a crafty tactic of redistributing their drafted auxiliary to distant parts of the empire. Britons and Gauls may well have served in Africa, Mesopotamia and Dacia, whilst Assyrians and Iberians along with many others, would have been some of the cultural background for soldiers serving along Hadrian's wall.
None of these men settled there. Roman tokens excavated in London do seem to confirm that some of the native girls would have earned a living as prostitutes, and that would have likely led to some undesired pregnancies. Yet as a whole, men who had served their term in the legion would be released as free men and given their pension, as well as a plot of land in their home land.
Just about the only large group that settled there to which I have ever been informed of during the Classical era, were a band of Sarmatian mercenaries that made their new homes in the region of modern day East Anglia. However I should state now that this is information I cannot seem to find a solid document to confirm, so it might be speculation. I'll look for where I found the information last time, but otherwise I'll remove this paragraph.
Moving later on to the period following the collapse of the Western Roman Empire and the period of history where the Angles, Saxons and Jutes made their crossing, there was a large diaspora of Bretons. They migrated to Amorica, which is modern day Brittany in France. Perhaps the connection there is quite clear. To this day, the Bretons still have some sense of Celtic identity that the Franks couldn't quite root out.