I actually have written about this topic a bit before: https://m.reddit.com/r/AskHistorians/comments/3jbpyk/britains_role_in_running_hannover/

The thing to remember is that there was a personal union, not a political union, although Hanover's foreign policy and government in exile were at different periods run out of London. Hanover was a possession of the crown, but had its own independent existence. The kings of Britain were only also kings of Hanover after 1814; before that they possessed it as electors.