After Georg Louis of Hannover ascended to the throne of Great Britain, how important was the German territory of Hannover to the British empire? For example during the 7 years war French troops besieged Göttingen and occupied Hannover as well as other hannoverian holdings on the continent. What kind of effect would this have had on the hannoverian king of Britain and the wider kingdom as a whole?
Edit: Britain in the title because I am dumb.
Very important in the early Hanoverian period (1714 - 1760). Indeed, it took until the 1760s (George III was the first Hanoverian monarch to have been born in England and to have spoken English as a first language) for the British to develop much affection for the Hanoverians since George I and George II were very German (George I made almost no effort to learn English at all), were often absentee and kept a very austere court in Britain. The British political elite anticipated all of this, having experienced what it was like to be ruled by a foreign, largely absentee monarch under William III in the 1690s, and so in 1701 created the Act of the Settlement, which limited the royal prerogative by forbidding kings from giving any lands, offices or titles in England to any foreigner and requiring the consent of parliament for any declaration of war in aid of Hanover. One could even potentially argue that the personal union with Hanover helped the development of the role of the prime minister, since these German kings were so often absent and regarded with suspicion by their subjects when present. From a foreign policy standpoint, it certainly helped drag Britain into the War of the Austrian Succession in 1740 - 1748.
In the late Hanoverian period (1760 - 1837), it arguably became less important, as George III, George IV and William IV, being born and raised in Britain, naturally took much greater interest in British domestic politics. The end of the union with Hanover in 1837 - Queen Victoria, being a woman, could not inherit the Kingdom of Hanover since it had a semi-salic law and so the throne passed to Ernest (the youngest and last surviving son of George III) - which was a relief to the British, yet its interesting to contemplate whether the UK would have tried to stop German re-unification if it was still in the personal union with Hanover in the 1860s.