What you must do is draw a distinction between the population of Britain and the amount of men it could feasibly enlist in the armed forces. You don't really start hitting population limit of soldiers in the armed forces until the 20th century, before then population was rarely the limiting factor, rather the amount of soldiers that could feasibly be paid, supplied and equipped. Remember that a soldier is deadweight as far as the economy is concerned, they produce nothing and cost to maintain, and every man that is the army or navy is one less in a field or factory contributing to the economy. For this reason a peacetime force in Britain was never anywhere near the population limit. Because the British empire covered a long period, il basically split it into three parts.

The 18th and early 19th century saw Britain in numerous wars, with Seven Years war and French revolution/Napoleonic being the most major. The size of the British army and navy during this time wasn't generally limited by population, mainly it was due to a huge difficulty in recruitment. The army in particular suffered somewhat from an image problem, and although the navy was better it wasn't exactly great either. Being a soldier was not attractive, the pay was bad, the life was hard and of course there was always the chance of death. The armed forces were therefore made up of people with no option than to join, if you had a chance of a job otherwise you probably wouldn't enlist. Gaps were filled by criminals being given service in the army as an option instead of imprisonment, and press gangs were used first by the navy and then the army. For most of these wars, the government was enlisting as many men as it possibly could, but as there was no official policy of conscription it could never get enough to join.

For maintaining the empire in the 19th century, the army in particular was really rather small. Its important to remember that Britain was a naval power. The Royal Navy was vast but it wasn't as manpower intensive as a large army was. One of the reasons the empire was successful was because it didn't need hundreds of thousands of troops everywhere to run it. It relied heavily on native troops in many places, for example the majority of British soldiers in India were Indian themselves. Think of the army during most of the 19th century was a sort of imperial police force. They could be deployed where needed to stop rebellion, but you didn't need a massve army for this. Really the empire was maintained for as little cost as possible.

The only time you really start hitting a population limit is in the two world wars. Total war was the first time that population ever really became a limiting factor for Britain and even then it would have been possible for the army size to be expanded further. About 5 million men passed through the army in WW1, and about 3.5 million in WW2. These are big numbers, but for example the the WW1 figure represented about 25% of the male population of the UK. Britain was never as desperate as say Germany was during either war. By the end of WW2 for example, Germany was forcing teenagers and old men into service, with over half of the German male population serving.

It should be noted that the world wars represent a totally unsustainable level of recruitment. In a way Britain did hit a sort of population limit. Keynes advised the post war Labour government to demoblise as quickly as possible because there was a labour shortage in the country and the men were needed back in the factories to begin rebuilding the economy.