Correction, Grey Wolf: Hannover WAS an independant state, it just happened to be in personal union with the UK. It was not a British colony. Now, if you say there was a Viceroy, well, thats news to me, but Ill trust you. Still, it wasnt a British Viceroy, but just the deputy of a King who happened to be away (in London, that is).
That difference isnt merely theoretical, either: The British King also rules in Hannover, but as an own office and title, as King of Hannover - and the British Parliament, though does of course not. Hannover has an own parliament, but as it was an absolutist state without much power. It also has an own administrtaion, an own government, etc etc etc. It jsut so happens that the head of that government, the monarch, is also monarch of a more senior realm, where he resides.
That being said, as long as the Personnal Union continues, the two (2!) realms will of course have an intertwinned history.Now if you mean with Prussian attempts to unify Germany Bismarck's policy (instead of, say, teh Erfurt Union of 1850)... well, I still maintain that wasnt really Bismarcks aim, but, oh well. One way or another, if unification coems from above (instead of form below, the people, as was tried 1848), Prussia and Austria will come to blows. With such an early PoD it might not be 1866, it might have totally different backgrounds, but a confrontation seems inevitable.
I think Hannover would stay neutral in such a war. Now, IOTL, Prussi apressured all neutral North german states (and its allis, too) into theNorth German Cofnfederation, but Hannover ha steh UK behind it and thus it cant be done with Hannover. And that would effectively shut out Hannover from the German unification process. So, a solution would be that as IOTL, Hannover sides with Austria and then get conquered and annexed by Prussia. Of course that would also mean the Uk sides with Austria, but naval power wont be required in that war, so I dont think much will change.
Another solution is the 1848 revolution. Now, her eliekwise it might not be 1848, but there will be democratic-nationalist unrests. IOTL, 1830 there were of some kind europe-wide, too, and then latr and bigegr again 1848, so I think due to the sentiments of the time there will be some nationalist-democratic unrest. Now, IOTL Prussia was the saviour of absolutism in Germany. Its troops intervened about everywhere and put the unrest down. This gave Prussia enough power to try to force a German Union (above mentioned "Erfurt Union", so named becaus ethe negotiations took place in Erfurt) on those states. The project was eventually ended because the states had no interst in such an union, Prussia became kidna esintersted, too, and eventualyl Austria returned to the German scene.
But lets imagine two alterations to that OTL happenstance:
1) Unrest is much, much worse in Hannover, so much that the King's Government becomes more or less unable to rule even after proclaiming a liberal constitution. Prussia intervenes as per OTL (I cant see Britain intervening to put down a revolt demanding a constitutional monarchy, I dont think Parliament would go along with it), but the Prussian presence is much more needed then IOTL. Hence the King of Hannover (who is also King of the UK) concedes that compromise with Prussia is necessary.
2) Prusisa puruses the Erfurt Union projet more eargerly, and for some reason Austria keeps gettingd sirtacted in Hungary so that, summarised, it comes to pas, including for aforementioned reason Hannover.
Reactions in Britain would be... intersting, surely. This would make their monarch a subordered monarch of a non-souvereign state within another country. They could feel that this will more or less compromitt their Monarchs position and integrity, and hence an act of Parliament (which after all controlls the UK laws of succession) might then force dynastical split on the King.