They did indeed have a centralised command structure, focused on the old Horse Guards buildings in London near St James Palace. The administrative heart of the growing British army had been based here largely from the early 18th century and would continue to do so well into the Victorian era until the then Commander-in-Chief the Duke of Cambridge was dragged kicking and screaming into the new offices in Pall Mall. (I exaggerate slightly, but he did have to be ordered to do so by the Queen, such was his reluctance). However, in the period you are interested in, yes the Secretary of War, and the Commander-in-Chief of the army was responsible for formalising and implementing the government's strategy for the army.

As for your perception as to the form of the British army in the period there is a 'colonel' of truth to it (kernel/colonel - get it? oh how I laugh). For a good portion of the English and late British army's existence, the regiment was the primary focus for administrative and raising purposes. Usually a wealthy man (often a noble although MPs and other wealthy gentry were known to) would be issued a commission to raise a regiment and he would do so. He would usually stand surety for the loans or front the cash necessary to recruit and equip the regiment, and be reimbursed at a profit by the government for doing so. By the later period you are interested in, the regimental system had become much more formalised, and while regiments still had their colonels (usually honorary) the constituent battalions of the regiment would be commanded by an experienced officer - the lieutenant colonel.

Purchases of commissions did happen, and remained so for well into the 19th century. However, this themselves went in through several stages of adjustment and change until in its final form, an officer had to serve a minimum length of time in rank before he was eligible to purchase the next.

If this topic interests you more, I can recommend reading about one of the unsung heroes of the British Army - Frederick, Duke of York. That is the man who as Commander-in-Chief of the British Army is responsible in turning the corrupt, inefficient, and informal British Army into the weapon that men like Arthur Wellesley could wield. He did this by standardising officer training and professionalising much of the officer corps limiting the amount of opportunists, profiteers, and amateur officers. He also formalised relations between militia and regular forces, and improved the army by supporting the growth of specialist units such as light infantry regiments, as well as improving pay and conditions for the regular soldier. Point of interest, so popular amongst the troops was the Duke of York in a period where old soldiers often opened or ran inns and pubs that to this day in England 'Duke of York' is a far more common pub name than 'Duke of Wellington'. Alas, if most people today have heard of him, it is usually through the nursery rhyme.

Title meant to say 18th/19th century. Sorry.