Find some way to make the Brits want to integrate with Hanover. That seems like the easiest route.
United Kingdom of Great Britain, Ireland, and Low Saxony perhaps? Give them Westphalia, Oldenburg, Hamburg, and semi-defensible borders on the Rhine, Elbe, and Central German Uplands and it could work out fairly decently. Just couple that with effective diplomacy that has Denmark-Holstein and the Low Countries in the British orbit/allied. Liverpool, Manchester, and the Ruhr in one polity? Oh my.
But anyways, focusing on the geopolitics, the British would likely focus on protecting their buffers/partners in the low countries and Denmark (which would likely be good for them because I can see them getting the benefit of the British trade network and British naval protection out of this). The British would be very similar to OTL Austria in that they'd be very very wary of German unification as well, at least by the major powers of Prussia and Austria. A Unification of the "Third Germany" outside of Britain (Saxo-Bavaria) might be something the British take an interest in however, as it would box out the Prussians and the Austrians.
Relations with the French are to be interesting as the British will want to protect the low countries from the French but would be friendly with the French with regards to not wanting a Prussian or Austrian unified Germany.
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Alternatively, you could just have the British accumulate enough sporadic territories that they feel a need to be tied to the continent. Ceuta and Melilla, Gibraltar, Menorca, Corsica, Malta, the Ionian Islands, and Heligoland could all feasibly be British due to the Napoleonic Wars and the period just before it.