Except a lot do just look at other countries in Britain, as part of "their" country. For instance, when the issue of political independence comes up, I've heard some English commenters say they hate the Scots for trying to split up "their country". No one's trying to split up England.
I'd never look at England as part of my country, because it isn't. We're in a political union with three other countries, but that doesn't make those countries part of mine.
And being part of a political union wouldn't shape what teams I support. When we were in the EU, I didn't preference EU teams over non-EU teams. I have always been neutral, outside of Scotland games. I have no ties to England, so I have no reason to root for them.
Ideally, I'd be neutral, but its that entitlement expressed above, and also the bias of the "British" media and "British" brands (chocolate bars, Walkers, Tescos, etc) and the main royals, like Charles and William, etc, that the English team, and only the England team, it's enough to put many of us off.
The same people who claim to be British, become very English all of a sudden.
I don't recall ever seeing the Welsh flag on a Mars bar, even when they qualified for the World Cup a few years ago.
As far as individuals go, I treat everyone with respect, regardless of their nationality, unless they're a-holes (Scottish, or non-Scottish).