It began with the letter G.
They used different systems to be sure - here are some H-class destroyers: Hardy, Hotspur, Hunter, Hostile, Havock (and others).
I'm not sure, but it certainly seems as though they had a destroyer class beginning with nearly every letter of the alphabet.
They also had the Tribal class - Eskimo, Punjabi, Cossack, Bedouin (and others). these were reportedly created as a response to the German destroyers, which were significantly larger than any then-existing class of British destroyer. The Tribals had eight 4.7-inch guns in dual mounts, as opposed to the typical single-mount configuration of the other classes in service in 1940.
Most classes were built with the intention that each class would comprise their own flotilla, and each class as a result had a slightly larger flotilla leader as a part of that class. For the H-class, for example, the Hardy was the flotilla leader - she displaced 1505 tons, while the other members of the class displaced 1340 tons.
As for cruisers, I believe they used city names.
Battleships and Battle Cruisers were named after Royal Navy heroes or royalty, such as Nelson, Hood, King George V, Prince of Wales, Queen Elizabeth - or abstract heroic concepts such as Repulse, Renown, Valiant, Warspite. There was often no naming pattern within a class, although Nelson and Rodney were both naval heroes. The Hood was originally part of a 4-ship class, but the Anson, Howe and Rodney were all cancelled (and again, all military heroes).
Carriers - well, here you can just go figure. They started out with flight references - Eagle and Hermes. After that they switched to Courageous and Glorious, so again, abstract heroic concepts. Hey - they had a lot of ships, so I would guess they sometimes had to scrounge around for names. Ark Royal was a traditional ship name - there had already been several RN ships of that name in the past. But then they went back to the Illustrious class, where it looks like they tried to use a single letter but ran out of steam: Illustrious, Indomitable, Implacable, Indefatigable, Victorious, Formidable. Stretching it a bit, eh?