If the kingdoms would have two seperate coronations it could still be sent to London for the English coronation.

Personally, I don't think the stone should be used in an English coronation. King Edward I of England took the stone as a symbol of his 'authority' over the Scots (He's not called the Hammer of the Scots for nothing) and subsequent English monarchs used it for similar reasons. It was only after 1603 that the stone and St Edward's throne could be said to be a symbol of unity.

If (and it's a big if) there are seperate coronations, one in each country, then the stone only needs to be used for the Scottish one. It has no real use in an English Coronation.