I think that's a generally accepted view of the Saxons, although there is evidence against it, and I remember one genetic study that found evidence of rampant Saxon settlement. Certainly East Anglia has the highest level of settlement though.
You're certainly correct about the proliferation of Scandinavian genes, particularly in the north - the west of the Pennines more than the east if I recall right. I don't know too much about studies of Scandinavian genes in Ireland, although I'd have thought Dublin and Cork/Wexford/around the Shannon would be the obvious areas. I think the most Scandinavian areas though are the north of Scotland/the Isles. I have read one study suggesting that red hair in the Celtic race is actually a sign of Norse heritage - and Scotland has more of it than anywhere else, and Ulster is the area of Ireland with most of it (Scots descent and all that). You can certainly imagine O'Connell leaving a longship looking for a church or two to burn... although you'd then imagine that Scandinavia would have more redheads than Scotland/Ireland, which it doesn't, so take that one with a pinch of salt obviously. Plus all the historical evidence towards redheads amongst the Celts anyway.
But you get signs of descent from everywhere... there's a family in Yorkshire who've been there for hundreds of years where all the men have a genetic expression normally only found in North Africa.