Simple answer - no. Scotland would be under the same crown as England, just like before the Union of Parliaments in 1707. Scotland would remain in the Commonwealth. The Queen is Queen of Australia, Canada, New Zealand, etc. However, if the people of Scotland decided in a referendum to abolish the monarchy in Scotland, then Scotland would become a republic. But there is no legitimate claimant to the throne of an independent Scotland and no-one has any Jacobite plans to instal one.