"Mrs Windsor's lines back to the first millenium, most of the names that come up are Anglo-Saxon."

May indeed be so. On the paternal side she will have a great many English and Scottish ancestors. She can also go right back through Robert the Bruce through his Gaelic mother and back to the early Kings of Scotland and even back to the Pictish kings certainly as far as Kenneth MacAlpin so basically to the mid-9thC. After that it gets more into myth rather than hard facts. Along the way there are Drummonds, Mures, Bruces, Dunbars as well as various known Gaelic ancestors from before the surname era. Add to that the fact that many of the early Scottish kings married English women then you bring in the Tudors, the Beauforts, and the House of Wessex itslef that was defeated by the Norman Conquest. It is true that one can't know all the fathers are who the tree says they are - but that is the same for everyone. So she has this elongated British family tree going back more than a thousand years on one side whilst on the other side it is basically a British lineage. Add to that the fact that the family have resided in Britain continuously for centuries then you can see it is absurd to say she isn't British.