There was an influx of “Germanic” folks, such as Hengist and Horsa who settled/ took a lot of the south. There were some skirmishes, I think a local tribal leader married Hengists daughter as part of a truce.
The “invasion” is largely over played, there is little archaeological evidence of large scale battles, it’s more a case of immigrants settling and language/ customs mixing although less “Celtic” language remains.
Another thing to consider is that still after the Roman Invasion, most areas were individual kingdoms, so there was little or no uniting to repel invaders.