Whose idea was it? Was there resistance to the idea from the royal family themselves, or traditionalists outside of the family? Was anti-German feeling in Britain really strong enough to force the Head of State to change his name?
Yes it really was that bad to be connected to Germany.
It even forced in part the resignation of the First Lord of the Admiralty. Admiral Prince Louis of Battenburg was Austrian by birth, and had been raised in Germany and joined the RN at 14.
He was a widely liked officer and generally very respected for his seamanship skills if not for particularly strong leadership.
But combined with age, his origins caused enough grief to have Churchill request his resignation. In order to get Fisher back of course. But the shift and causes were lamented through the service as he had overseen the fleet readying to a war footing.
His son then was the famous Mountbatten of Burma after a name change and his grandson is Prince Phillip the Royal Consort.
The anti-German feeling all over the British Isles and the United Kingdom caused by the war was indeed strong enough to persuade King George V to rename his own German family/dynasty name of "Saxe-Coburg and Gotha" to "Windsor". Furthermore he forfeited all German titles by request of his British relatives.