I'm going to answer this with reference to HMS Hood, as I have exact numbers for her crew available. Hood had a larger crew than the Queen Elizabeths, and the proportions of the crew are not exactly the same. Both had the same primary armament, but the requirements of their secondary armament and engines were very different. The numbers I have are also for Hood in 1919, and do not reflect changes as new technologies like additional anti-aircraft armament or radars were added. You might also be interested in my answer on the officers of the King George V class here.

Hood's crew was, like that of every other ship in the RN, divided in three ways: by department, by division, and by watch. Speaking roughly, these represented what a sailor did, where they worked, and when respectively. The departments were broad collections of officers and ratings who worked on a particular area. The largest department, the Executive Department, was broken up into a number of branches with greater focus, such as gunnery. Capital ships had nine divisions, with the crew assigned to these by where they worked or their speciality. The ship was broken up into four equal parts, and seamen without specialities were assigned to them. Each of these divisions was responsible for keeping their part of the ship cleaned and maintained, as well as manning it in action. The stokers in the engine room had their own division, as did the gunnery, communications and torpedo specialists. There were also divisions for the Royal Marines and for 'Daymen', who worked outside the watch system. Battleships were typically run on a two-watch system, with watches called 'port' and 'starboard'. Under the two-watch system, the men would spend four hours on duty, and four hours off. The two watches were also subdivided into two parts. These were important when the ship was in harbour, when only a single part of a watch would stand duty. The engine room was run on a three-watch system, with the stokers divided into three groups. This allowed each watch to have eight hours off duty uninterrupted. As noted above, there were also daymen, such as the medical and supply department. These worked during the day, and did not stand watches.

The largest department was the Executive Department. On Hood, this had 710 ratings and 32 officers. It was responsible for much of the duty of fighting and maintaining the ship. It was subdivided into several branches. The seaman branch was the largest, with some 379 men and 29 officers. As might be expected, it did most of the work around the ship that did not require a specialist, such as painting, cleaning the deck, and watching for aircraft and other ships. Seamen also helped man the turrets, doing the menial work. The gunnery branch was the next largest, with 246 men and an officer. These were gunnery specialists, who did the majority of the work in the turrets and on the secondary armament, as well as in the fire-control director. The torpedo branch was, as the name suggests, responsible for the torpedo armament. By historical accident, it was also responsible for the ship's low-power electrical systems. Hood's torpedo branch had one officer and 85 men.

The Engineering Department was the next largest. Their job was to run and maintain the ship's engines, as well as its hydraulics and high-powered electrical systems. Hood's Engineering Department consisted of ten officers and 293 ratings. Most of the ratings were stokers, but 39 were the more highly trained artificers and mechanicians. Stokers did the majority of the work that needed doing, while the artificers were more able to make repairs and carry out maintenance. Not all of the stokers were required when the ship was in action. As a result, about a quarter of them would form damage control parties and help man the armament. The ship's Royal Marine detachment consisted of 181 enlisted men, under the command of a major and two lieutenants. There was also a 17-strong Marine Band. The Marines provided the ship's butchers, and acted as sentries. They could be landed to fight ashore, though this was rare. Hood's marines would fight in Norway in April 1940 while the ship was undergoing a refit, in one of the few cases I know of of this happening. Most of the time, the Marines manned the ship's aftmost turret. The last of the watch-standing departments was the Communications Department. Composed almost entirely of ratings, it had 50 men on Hood, plus two warrant officers. As you might expect, it was responsible for the ship's communications. There were thirty signalmen, who sent and received flag signals, semaphore and light signals. There were also twenty telegraphists, who operated the ship's radios.

The remaining departments were all daymen. The Supply Department had a broad range of duties. As well as looking after the ship's supplies, it was responsible for cooking food (though only for officers on ships older than Hood), doing the ship's accounts and other paperwork. It also provided stewards for the officers. Hood's had four officers, a warrant officer and 34 ratings. Most of the ratings were cooks and stewards for the officers. The Medical Department was small. It had three officers, all medically trained. One might be a dentist. They were assisted by six ratings. Three were stewards, who did cooking and cleaning for the invalids or wounded, while the other three were Sick Berth Attendants. These performed a variety of duties, acting mainly as nurses and assistants to the surgeons, but could also carry out first aid. The last formal department were Artisans and Artificers. This was a varied collection of skilled mechanics. They did repairs and maintenance that could not be carried out by the less-skilled men of the other departments. Hood had 28 artificers focused on the maintenance of the ship's hull, deck and fittings, including three plumbers, three joiners and twelve shipwrights. The armament was looked after by twelve Ordnance Artificers. There were also fifteen electricians. Finally, there were a number of other miscellaneous men aboard. This included the ship's chaplain, its police (six men from the Regulating Branch), a physical trainer and six divers.