No interest in the replies from John Simmonds?

I thought that the British were generally pushed west and north by the Celts when they came to what we call the British Isles about 3 centuries before the Romans turned up.

No, the British (Pritani, Brettoni etc.) were the "Celts" who were there when the Romans arrived, and also after they left. (What Celt meant, apart from language, is a matter for debate.) They may have displaced an earlier population, and they may have merged with them, or there may simply have been a change of culture over some hundreds of years. DNA evidence suggests both continuity and immigration.

The English got their name from the Angles, but it doesn't mean that every Englishman was an Angle- the probability is that the immigrants mixed with the indigenous population in varying degrees. Then of course Norses of various flavours came along and added to the Micks (sorry, mix) and changed the language again. And so on.