Many of us remember Kaiser Wilhelm II's famous joke about looking forward to seeing a performance of Shakespeare's play, The Merry Wives of Saxe-Coburg and Gotha, but I'm curious about how this change (and the corresponding change of the name in Belgium as well) was greeted by those German members of the family who still maintained it.
It's perhaps helpful to put the changing of names into context because the whole thing was part of a wider series of tit-for-tat glove-slapping that was confused not just by the fact that Britain's aristocracy was riddled with Germanic transplants, but so was Germany's awash with Brits.
The head of the House - the British-born grandson of Queen Victoria, Charles Edward, Duke of Saxe-Coburg and Gotha, better known as "Charlie Coburg" - remained sympathetic to his English relations and kept correspondence open with his sister, Princess Alice.
There's reports that he refused to serve on the Western Front out of familial bond - but then other historians describe him as a "chocolate soldier" who simply wined and dined in his lovely uniform, so it's unlikely anyone was expecting him to go the front in the first place.
For much of his pre-War life he remained torn between German and British roots - he kept Scotch terriers and was criticised for his "foreign" accent. He was even in England when Archduke Franz Ferdinand was shot. (By the 1930s he'd hardened in his views enough to join the Nazi Party, but that's an aside.)
That Kaiser Wilhelm II was his mentor similarly left him torn, as Willy himself was seen as being "too English" and had occasionally lurched into full-bore Anglophilia depending on which phase his troubled love/hate/love relationship with his cousin (George V), uncle (Edward VII) and his grandmother (Queen Victoria) was in at any given time. Wilhelm was even much admired in Britain for the statesmanlike way he had comported himself at Queen Victoria's funeral.
This left Charlie in a pretty sensitive place during World War I, and he concentrated on reinforcing the German credentials of Saxe-Coburg and Gotha. He officially renounced his ceremonial rank in the Seaforth Highlanders ("I can not be in charge of a regiment... which belongs to a country which has attacked us in the most despicable way.") and in March 1917 he signed a bill to strip his British, Belgian and Portuguese relations of their German titles. This included his sister, Princess Alice, and his mother, Princess Helena.
Back in Britain anti-German hysteria was churning up a storm and Admiral Prince Louis von Battenberg (married to one of Victoria's seemingly endless parade of granddaughters and father of the future Louis Mountbatten, larger than life Earl of Burma and Viceroy of India) was hounded by the popular press and forced to resign in 1914 as despite being a naturalised Brit, he was German by birth.
On 25 May 1917, 23 German aircraft bombed the South of England leaving over 100 people dead and over 200 wounded. The model was a Gotha G.IV heavy bomber.
The British branch of Saxe-Coburg and Gotha changed their names in July 1917 and the British Titles Deprivation Act 1917, which stripped the German nobles of their British titles, was passed in November that year (stripping Charlie of "Duke of Albany").
According to Royal Babylon: The Alarming History of European Royalty by Karl Shaw the German branch of the House of Saxe-Coburg and Gotha (as well as the House of von Battenberg and House of Teck) was "scandalised", with many "minor" members of British branch similarly less than impressed by this betrayal of their ancient birthrights. Alexander Cambridge, 1st Earl of Athlone - formerly Prince of Teck - doesn't seem to have been a willing participant, with his wife (the aforementioned Princess Alice) recalling him "furious, as he thought that kind of camouflage stupid and petty."
The Bavarian count (or "Graf", which I enjoy using because it makes me think of Nosferatu) Albrecht von Montgelas commented, "The royal tradition died on that day... when, for a mere war, George V changed his name."
Sources: Go-Betweens for Hitler by Karina Urbach and Royal Babylon: The Alarming History of European Royalty by Karl Shaw