How did the English nobility react to the removal of the German Monarchy in 1918?

In short, they were elated.

Contextually, Wilhelm II's reign ushered in an era of unexpected and aggressive expansionism even by German standards. Wilhelm's father, Frederick III, was known as a compassionate and liberal monarch and his succession to the throne was expected to herald in an era of competent politically centrist rule. He was particularly popular in Great Britain due to his marriage to the Princess Royal although his Anglophile views sometimes drew some domestic consternation by members of the German government. Unfortunately, Frederick III died from throat cancer and lasted less than a year on the throne. Wilhelm was unlike his liberal father in every way. Despite the best intentions of his parents, Wilhelm fell under the tutelage of a series of junkers who proceeded to drill Prussian militarism deep into his brain. A series of rather dramatic faux pas alienated British opinion against him. What follows is a very short list of international incidents that absolutely enraged the British Foreign Office between 1896 and 1914:

Wilhelm sends the Kreuger Telegram congratulating the Boers on repelling a British assault (1896)
Wilhelm sparks a naval arms race with Britain by commissioning Admiral von Tirpitz to modernize and dramatically enlarge the German High Seas Fleet, thereby infuriating the Royal Navy (1897)
Wilhelm holds a reception for Boer generals at the Neues Palais in Potsdam immediately preceeding the visit of his uncle Edward VII
Wilhelm gives an interview to the Daily Telegraph and proceeds to outrightly insult Britain in every way possible (1906)
Wilhelm throws a tantrum in front of Prince Louis of Battenberg, subsequently claiming that Germany was the sole power in charge of European continental affairs. This was all, mind you, while on the deck of a British battleship (1911)

I mean the list really goes on. Things had gotten so bad that the First Sea Lord called for the Royal Navy to preemptively sink the German fleet while in dock at Kiel as early as 1908. (Side note: A great source of "Did he really say that?" material comes from "Wilhelm II: Into the Abyss of War and Exile, 1900–1941" by Röhl, whose book on Wilhelm is quite authoritative.)

By 1916, there was absolutely no question that Wilhelm's abdication would be an eventual condition of any Allied victory. His cousin, George V, was so thoroughly disgusted with Wilhelm that he changed his last name from "Saxe-Coburg-Gotha" to "Windsor" and the Russian Emperor had an even larger axe to grind as a result of Wilhelm's earlier agitation for Russia to enter into a disastrous war with Japan. However, this doesn't exactly address the removal of the German monarchy per se. While there were private calls for Wilhelm's abdication, absolutely no one had considered the idea that Germany would become a republic.

In the weeks preceding Wilhelm's abdication, Germany became possessed by a series of extremely unstable socialist revolutionary movements that sought the proclamation of a republican form of government. This was anathema to an enormous segment of the German population, which remained inherently conservative and monarchist at heart. Prince Max von Baden, the Chancellor, attempted to prevent the abolition of the monarchy although he was ultimately thwarted by Frederich Ebert, who panicked and ultimately proclaimed a republic to in order to please demonstrators to von Baden's horror. I would say that any permanent republican system in Germany was probably unforeseeable by the Allies in 1918, although the English nobility believed that their cause was just and that the antebellum government in Germany deserved some sort of revolution based on prior actions executed in the decades prior to the First World War.