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The first King of the Belgians, Leopold I, then Prince Leopold of Saxe-Coburg-Saalfeld, married in 1816 Princess Charlotte, the only child of the Prince Regent. Tragically, just 18 months later, Charlotte died in childbirth. As a sort of compensation for lost opportunities, Prince Leopold was granted a very large annuity by the Parliament.
In 1830 Prince Leopold was invited by the Belgians to be their King after they had broken away from Holland. As the uncle of Queen Victoria, who was his sister's daughter, he encouraged the belief that he could influence British policy. In fact, his influence over his niece was not very great, fond though she was of him and much as she valued his political advice. His position with regard to Britain appeared to be even stronger after Victoria's marriage to Prince Albert, one of his Coburg nephews.
The Saxe-Coburgs of Belgium by and large shared the majority religion of the Belgian people, Roman Catholicism, ruling out marriages between the British and Belgian Royal Houses. After Leopold's death relations between the two Houses cooled as Leopold II was the least attractive European monarch of his day and headed a very dysfunctional family. King Albert I was a respected wartime leader during WWI but his successor Leopold III undermined a lot of goodwill by his 2nd marriage during WW2 and (unfounded) accusations of collaboration with the Nazis.
Relations between Elizabeth II and Albert II are no doubt warm and cordial but there is no longer any particular link between the two Houses apart from their common Saxe-Coburg ancestry.