The books suggested by /u/baetawolf are fairly old and don't take into account recent evidence, particular in terms of archaeology. I'll talk about three books published in the last decade that should provide you with a decent overview of this period, as well as being fairly readable, as they were all written with a popular audience in mind. Robin Fleming's Britain after Rome: The Fall and Rise, 400 - 1070 is a very accessible new book covering this exact issue, favouring archaeological evidence over literary evidence (which makes sense, we literally only have the highly problematic polemic of Gildas to inform us about the arrival of the Anglo-Saxons). It essentially argued that the Anglo-Saxons did not flood the island, but that only a small number arrived, though in the process they profoundly changed native culture.

Another useful introduction might be Guy Halsall's Worlds of Arthur, which dealt not only with the origins of King Arthur (spoiler alert, he wasn't a real historical figure) but also proposed a highly innovative thesis - that the Anglo-Saxons were invited into Britannia as auxiliary forces by the usurper Magnus Maximus in the late fourth century. Once Roman power truly collapsed, these soldiers remade themselves into the masters of the province. It is a very fun read, though you should take his argument with a grain of salt, but it nonetheless demonstrates very well how the traditional narrative is fatally flawed.

For a more traditional take, Peter Heather's Empires and Barbarians: Migration, Development and the Birth of Europe's section the Anglo-Saxons is a good place to start, though again he did not emphasise mass immigration, instead arguing for a transfer of power from a small Romano-British elite to a larger Anglo-Saxon one. Still, even according to Heather the scale of population transfer was fairly small, though larger than the number of Normans who arrived after 1066.

In conclusion, I think these three historians would all agree with elements of the documentary, though as you can probably gather this is a contentious issue in early medieval history due to the lack of evidence, so we will most likely never reach a firm conclusion about the size or the impact of the Anglo-Saxon migrations.

You might need to explain what you mean a bit more. Is the challenge that the Angles/Saxons/Jutes never arrived at all, or an argument over whether their settlement constituted an invasion (rather than, say, forces being invited into Britain to protect against other raiders)?

For more information on the topic in general, I can recommend 'Anglo-Saxon England' by Peter Stenton, or better yet, 'An Introduction to Anglo-Saxon England' by Peter Hunter Blair.