In the 17th Century the Kingdoms of England, Scotland and Ireland were ruled by the Scottish House of Stuart. This was a period of friction between the monarchy and Parliament, including the English Civil War and the Glorious Revolution, the outcome of which was that we retained a monarchy, but Parliament was firmly in charge of who the monarch was.

And Parliament did not want a Catholic King, indeed most of the point of the Glorious Revolution was to depose the Catholic King James II&VII and install his daughter Mary, who was a Protestant, as Queen, by inviting her husband William of Orange (also a Protestant) to invade England from the Netherlands.

This was okay as it went, but then William and Mary died childless, and the throne went to Mary's sister Anne, who also didn't have children. The next few Stuarts in line were Catholics, and Parliament really didn't want that, so they issued the Act of Settlement 1701, which decreed that the throne would bypass the Catholics and go to the next person in line who was a Protestant, which happened to be Sophia of Hanover, daughter of Frederick V, Elector Palatine and Elizabeth Stuart, who was the daughter of James VI&I of Scotland and England.

George I of Great Britain

Sophia died two months before Anne did, so the claim passed to her eldest son, George, Elector of Hanover, who became George I of Great Britain in 1714. (England and Scotland had been unified as Great Britain by the Acts of Union 1707).

So George became the first Hanoverian King of Britain, because his great-grandfather was a Stuart King of England and Scotland, and the British Parliament considered it more important to have a Protestant monarch than to have one from Britain.
The important thing to understand is that when it comes to European royalty, none of this is particularly unusual. All the royal dynasties strategically intermarry and thrones pass from family to family as Houses extend or reduce their influence. It means nothing to Britain or British heritage.

George III (great-grandson of George I) was the third of the Hanoverian Kings and the first to be born in Britain and to speak English as his first language, indeed he never even went to Hanover. The monarchs after him were called Hanover but were definitely more British than German, culminating in the last Hanoverian monarch, Queen Victoria. Victoria, as a woman, could not inherit the throne of Hanover, so the connection between Hanover and the UK was severed here.

The British Royal Family in 1880.

Victoria may not have been German but her husband Albert of Saxe-Coburg and Gotha was (remember what I said about European royalty marrying each other all the time?) so the British royal family got another injection of German blood, and a change of name.
The House of Saxe-Coburg and Gotha had two British Kings, Edward VII and George V, before it was considered prudent to change the name because of anti-German sentiment among the public in WW1. So in the UK they became the House of Windsor.

George V, who was King during WW1, had no particular sympathies with the Germans, even though Kaiser Wilhelm II was his cousin. The fact he had some German ancestry would have made little difference to his dealings with Germany during the war.

Edward VIII actually served in the British army during WW1, and saw the front-line first hand, although obviously as heir to the throne he was kept out of immediate harm. He was only King for a matter of months before his abdication in 1936, but in 1937 he visited Germany, meeting with Hitler, giving Nazi salutes and generally being quite friendly to the Nazis. When WW2 started this was somewhat unhelpful, so he was made Governor of the Bahamas, in order to make sure he was a long way from Europe.

George VI, who became King when his brother Edward abdicated, did not share Edward's Nazi sympathies, and neither was he particularly German. His mother was a princess of Teck, which is in Germany, but she was born and raised in England.

In any case, his partial German ancestry was not considered an issue in official dealings with Germany, before, during or after World War 2. And neither is that of his daughter Elizabeth II, the current monarch.