It would all depend on the future relationship. If there was a common defence policy then why not? It is the Royal Navy, not the UK navy or the English navy.
As for what we will all be called, well, don't forget the separate country of Wales was conquered by England in the late 13th Century and the Laws in Wales Acts 1535 and 1542 fully incorporated Wales into the Kingdom of England. When Elizabeth died without an heir in 1603 England begged James VI of Scotland to also become King of England in a personal union. In 1706/07 the Acts of Union formally joined the two nations into the Kingdom of Great Britain, with a common Parliament and a common currency but separate legal systems. The currency, Parliament, armed forces and other assets and liabilities are jointly owned. It was a partnership of equals that over time, due to population and the concentration of power has become dominated so much by England that this is overlooked. So if this Union is dissolved assets and liabilities will have to be divided, but Scotland may well be just as entitled to use the pound.
We became the United Kingdom of Great Britain and Ireland in 1801 when the Parliaments of Great Britain and Ireland passed an Act of Union. Once Southern Ireland became independent in the first half of the 20th Century we became the United Kingdom of Great Britain and Northern Ireland.
So, if Scotland goes Independent would we still be able to use Great Britain? If Northern Ireland go too, can we even call ourselves a United Kingdom? Wales might complain about not being mentioned anywhere in the name as in pre 1707 times but that technically might be the result, we just become England.
I wonder if this would mean we will lose our seat on the UN Security Council and the G7? Isn't Brexit great?