I have a copy of General Regulations and Orders for the Army issued on 12 August 1811, to which are added such Regulations as have been issued to 1 January 1816. I cannot see any establishments in this, or perhaps I have missed something.
I have an idea establishments were published as separate documents and I note that J A Houlding, in his "Fit for Service: The Training of the British Army 1715-1795" lists amongst his references a document in the Public Records Office as WO 24 Establishments. I do not however have a copy of this.
There were variations as regiments were formed or enlarged at different times and establishments did not apply to all simultaneously. In UK, then and now, establishments were driven by money, ie how much the Treasury was prepared to spend, and therefore the Army frequently accepted a compromise of less than it really wanted, or possibly pragmatically accepted some regiments on higher establishments than others, so at least some were what it wanted.
All expenditure on the armed forces had to be approved by an Act of Parliament annually. If Parliament did not pass the Army Act each year then the Army had no right to exist (ie the default was NO army). This check was created after the English Civil War to prevent a King raising an army against Parliament's wishes. Establishments could therefore vary from year to year depending on what was approved by Parliament. This system was only changed in the last few years of the 20th Century.
In the end establishments only show what is authorised, and therefore what will be paid for. It was, and still is, up to individual regiments to recruit both officers and soldiers to fill those establishments, which they did with varying success, depending on the popularity of the regiment.
We still have vestiges of this same battle today, where many infantry regiments are single battalions with no ability of the Army heirachy to cross-post to remove variations in recruiting. The modernists want to amalgamate these into multi-battalion regiments to allow cross-posting, the traditionalists want to keep the existing system, which is perceived as retaining regimental traditions, and thence regimental "esprit de corps". In Scotland, for example, there are now six separate single battalion regiments. There is a proposal to merge these into a single five battalion regiment to allow cross-posting of both officers and men. Quite apart from the reduction of one battalion there is a fear of loss of regimental identity, and regiments like the Black Watch are fighting what I regret will be a losing battle against this proposal.