Personally I believe that if Hanover remains in personal union with Britain it would not get involved in the "unification wars". I don't think British parliament would want to get involved in those wars and thus force the king to remain neutral. Prussia (or any other German power) would fear British involvement and simply leave Hanover alone.I think that in the end Hanover will turn into a big Luxemburg. A "German" country in personal union with another country and thus staying outside of German unification.
An independent Hanover will probably change the unification wars" significantly, but for fun lets assume it doesn't, including the assumption that Prussia still manages to defeat France in the Franco-Prussian war even without the Hanoverian territory. First colonisation will probably be very different. Germany will not have a good port at the north sea, not even Hamburg since that is bordering British Hanover. I am not so sure this Germany would go for colonies. That would change the Anglo-German relations quite a lot. With no colonies there will be no naval race. No colonies would also significantly change WWI, since it would only be fought in Europe.So not much of a WWI. If it means Anglo-Prussian relations are still pretty good because of the lack of the naval race and the fact that Britain and Germany sort of border each other, the Franco-British alliance would be significantly weakened. Even without Belgium Britain was already pro-French. Belgium was a great reason to actualy join the war. If Anglo-German relations are still pretty good at the start of WWI, Belgium might not be enough reason for the British to join the war. France knowing that they can't count on British support might because of that not even declare war on Germany but look for a diplomatic solution thus avoiding the entire world war or simply turn it into a localised conflict. Actualy even if France joins the war, without Britain (and thus the USA) or any fighting in the colonies, this would just turn out to be a major European war, but not into a World War.