How would a "rump UK" be named after Scotland and Northern Ireland have seceded?
"United Kingdom of England and Wales"?
I suppose they would like to retain the "United Kingdom" moniker to display continuity and avoid overly elaborate renaming procedures. On the other hand, still calling the country "Great Britain" could be considered a bit preposterous, offend the Scots, and even lead to a new Macedonia-style naming controversy.
The Disunited Kingdom of Lesser Britain and No Ireland 
Why would Brittany be involved?
a return to the pre-1801 name of the Kingdom of Great Britain makes the most sense if they get rid of just Northern Ireland. If both NI and Scotland are gone, and they want to keep the UK name, I suppose that they could use the United Kingdom of England and Wales instead of just the Kingdom of England.
They could also reignite the Edwardian claims and become the United Kingdom of England and France.
