I was watching a video (https://www.youtube.com/watch?v=rexKqvgPVuA) and it discussed the history of English. Of course the Romans held control of the land now called Great Britain for the first few centuries CE, but left once the western half of the empire declined. They left parts of Latin behind and new settlers brought with them words that have made their way into the english language.
At what point did the language we now call english become english? Was it after the Anglo-Saxan migration or was it much later?
It depends a lot what you mean by "the language we now call english." Old English is what Beowulf is written in. Middle English is what Chaucer wrote in, but neither of these are really intelligible to a modern audience. Each of these are what we call english, but a different version of it. I am not sure if you want the birth of modern English or of the very first thing we call english which was a very different language but evolved into modern english over time.
Regarding what you said about the developments in England, Roman culture did not survive particularly well in Britain. The real influx of Latin cam in 1066 when the French (it was a proto-french, but whatever) speaking Normans invaded. It combined with the Germanic language that was there already due to the Saxon migration you spoke of and meaningful interaction with Scandinavians who spoke a different set of Germanic languages. The link above speaks about this some, but most of what you want it sounds like could be found on the wikipedia page