It was used in a number of other countries in Europe (Germany, France, Spain) but I'm not aware of any other countries - although it may have been used in the colonial armies of these countries as well. EDIT: I checked, and it was used in the (Honourable) East India Company's armies. The HEIC had its own armies in India, which were essentially private armies.
From what I read of British literature of the time, the rank and file soldiers get discharged and go back to being farmers and shopkeepers and so on. The officers (the ones who are permanently in the military) go back to their estates and their dinners and their fox hunting and everyone calls them "General" and "Colonel" and so on. The government isn't going to pay to keep them in a camp if there's no actual war going on. If war does start again, these officers will be summoned to the aid of the Crown. They'll likely be required to raise a regiment of soldiers from the neighborhood, which they will command.
If memory serves, it also depends on which branch of the service you're referring to. From Geoffrey Regan's Naval Blunders: The "Price" that Regan refers to is Rear-Admiral David Price, who -- after spending just four years at sea, four years employed ashore and the rest of his career unemployed and on half-pay -- was suddenly put in command of a squadron at Petropavlovsk in 1854. It was his first command in his entire life. Unable to cope with his responsibilities or his fears, he took to his cabin and shot himself. He even apologised to his officers for his "crime", telling them that 'he could not bear the thought of taking so many noble and gallant fellows into action'. Price was psychologically unfit for command, but he knew he was not up to the job and was looking for a way out a situation he could not resolve. What we would say now is not what they said then. Clearly Price was suffering from depression in an age that had no name for it except dereliction of duty.
I suppose this could have been the cause in earlier times or even just after the Napoleonic Wars, but was it really the case at the turn of the century between the 19th and 20th century? I thought the British had adopted a system of a small but fully professional army at that point. But I could be entirely wrong. Interesting with a sad, tragic and horrible example. I can't imagine what stress Price must haven felt to be thrown from nothing into a situation with so many lives depending on his decisions, just like that.
It was my understanding that WW1 was what killed off the old system of seniority with the death of most of the upper classes that traditionally fulfilled those roles. In the US, after WW1, they introduced rules for officers requiring mandatory retirement after being passed over a few times and regular fitness reports. I think. Might have been after WW2.