What would it take to extend the English crowns 18th Century possesions in the German states into a larger and more permanent realm of the empire? Large and solid enough it forestalls or changes the character of the late 19th Century German empire? Is eventually a British Dominion on the continent possible? Or even marriage bringing the British king to directly rule in a German kingdom?
Well Hanover was never really a part of the British Empire, however complex organisationally that actually was, simply in personal union and the British government generally considered it to be a bloody nuisance and would have liked to of been shot of it so I can't ever see it being added officially. Even becoming a dominion is likely impossible since you would need to find some way of derailing German nationalism and popular feeling for unification, even then I'm guessing Hanover would of had more trade opportunities with Germany that the UK which is another strike against the idea. That being said there are a number of things that could happen with it that have the potential to radically change how Germany develops.
Slight digression for the moment though. If you just want the British Empire holding territory on the continent then the easiest solution is to have Britain claim/somehow wind up regaining the Pale of Calais during the dividing up of Europe after the Napoleonic wars. IIRC it used to run roughly from Cape Wissant south-east down towards Marquise then north-east-east towards Audruicq before turning north-north-east-ish to hit the coast two thirds of the way to Gravelines. That would give you an exclave about twenty miles long and six miles deep.
But on to Germany! The first step is to avoid severing the personal union between Hanover and Britain that occurred when Victoria became Queen. Her uncle instituted a new rather liberal constitution in Hanover four years before he died, simple enough to have him decide that if his niece is good enough to rule the British Empire she's good enough to rule a German kingdom and change the succession laws at the same time. Come the Austro-Prussian War in 1866 the Battle of Koniggratz, which was much closer that most people expect, goes the Austrian's way and they win the war. During the peace negotiations they annex a number of the surrounding nations that sides with Prussia during the war - Bremen, Hamburg, Lippe-Detmold, Prussian territory north of the Ems river, potentially Oldenburg etc. - and Prussia forfeits their territory in the west and central Germany. IIRC they either resign from or denounced the German Confederation right before launching the war so I could easily see them being excluded from it and the
Zollverein for a good five or ten years before being let back in. All of which adds up to them being fairly unlikely to lead any future unification of Germany.
Then you just need to scheme to somehow get Hanover to lead any future unification. One crazy idea I had was for a Frankfurt Parliament that tries to unify Germany by excluding Austria and this smaller Prussia and asking Queen Victoria to be head of state in a purely constitutional monarchy position. She has the prestige, is more likely to be interested in the British Empire so won't try and influence things in Germany too much, her husband Prince Albert is suitable German etc. The other half of the deal is that the future Edward VII is required to renounce his position as heir to Hanover in favour of her second son Alfred who by this point was already set to become Duke of Saxe-Coburg and Gotha as the current ruler Ernest II was pretty unlikely to be having any children by that point. Victoria becomes Empress of Germany in the purely ceremonial role only travelling there once or twice a year for official purposes and on her death in 1901 Alfred becomes King of Hanover, Duke of Saxe-Coburg and Gotha and most importantly of all Emperor of Germany. Due to his future position he probably won't have joined the Royal Navy but instead be spending a lot of time in Hanover and Germany instead in preparation. Since I'm guessing that this new German Empire would be more of a federation than unitary state I wouldn't be surprised if Prussia had joined it at some point in the intervening period.