The formation of a Siege Battery is described in The History of the 135th Siege Battery RGA in May 1916.

The starting point was a letter issued by the War Office(3rd May 1916), in this case to the Commander Royal Artillery Tynemouth Garrison. This was in fact a Coastal Defence unit.

The initial cadre consisted of a Battery Sergeant Major, a Battery Quartermaster Sergeant and two Officers. The gunners in this instance were to be drawn from the 47th and 12th Company's RGA, both regular army coastal defence units manning the Tyne guns, and an inspection / selection process commenced.

The next expansion was the arrival of 10 NCO.s from the RGA depot at Bexhill, as well as some signalers. This would give the framework for further expansion, as well as provide experience as some of the NCOs had already served in France. Now the gunners started to arrive, 66 from 12th Company RGA and 44 from 47th Company RGA.

So within a week the 135th Siege Battery could commerce training, which was conducted in the Tynemouth area. The Battery spent the greater part of the time doing foot drill, physical training, and classes were held for the observers, BC's assistants, and signalers. By the 4th June the Garrison Band was playing farewell to the 135th, the Officers and men traveling overnight to Horsham in Sussex.

The Battery spent a month at Horsham learning technical skills, gun drill, laying, director drills. and signaling. Additional Officers joined the Battery to bring it up to it's establishment for Officers. On the 4th July they moved to Lydd to conduct live firing training. Thiat period lasted two weeks after which they were deemed ready to be deployed to a Theater

On the 29th July, half the Battery were sent on mobilisation leave, the remainder being sent to Woolwich to pick up gun stores and other equipment. The Battery then started to assemble near Bristol. It was here that the Battery's four 8-inch Howitzers arrived. Preparations were made to deploy to France. Men were promoted to fill the NCO's establishment, and the artificers (Staff sergeant, a wheeler, and a fitter) joined.

The advance party left for France 11th August 1916. The Battery deployed 13th August 1916, a little over 3 months since the letter from the War Office.

I would think the formation of many of the Batteries would have been similar as the Coastal Defence units provided a nucleus of experience men, and a framework. Where the men were drawn from would depend. The Territoirials of the Tynemouth and Durham RGA were drawn upon to provide the cadre for the 94th Siege Battery, the majority of the men were drawn from the New Army recruits.

Source: Forming a siege battery