I hear a lot about the abysmally small size of the British Army throughout history since its foundation. It is widely concluded it was more logical for the British to invest more into a naval force than a land force for obvious reasons. In the Napoleonic Wars, Wellington's victorious army in Spain wasn't even "British" in the sense that most of the army was made up of Spanish and Portuguese. The great victory of Waterloo was won by a "British" army of 68,000, of which only 25,000 were British.
I have two questions-
Yes. Almost every single military campaign that the English army fought in Europe was with a coalition of other forces. Partly it was due to a smaller population, although compared to many European states England's wasn't that bad. The primary reason was it lacked the willingness to mobilise large numbers of it's subjects to army service. "British" armies also had a long history of raising foreign units to serve in the army.
This is probably the reasoning for the references here to "professional". It's simply a different use of the word - a profession as in a job rather than in terms of an elite army. And let's also be clear, at various points in its history the British army has been quite shocking and varying in quality. But, unlike many European powers Britain first introduced conscription in 1916, halfway through WW1, whereas by then it had been standard in Europe for over a century. The "professional" likely refers to the use of the word to mean "regular" army.
Most of the soldiers in the other armies you list were not professional soldiers, but compulsory enlisted for the duration. Even in wars where manpower was at a premium, such as the Napoleonic Wars, British recruitment quotas were more often for manning home-service militia or volunteer units, and instead the army relied on volunteers, bounty men, or enlistment of felons. These were used rather than imposing conscription for the army (press for the navy was a bit different and isn't quite as popular imagination has it).
So to your comparisons with French or Prussian "professionals" you are fundamentally correct in that they were very different systems, mostly based on levels of preorganized and systematic applied national conscription, against a small "volunteer"-only enlisted regular force. (Stretching the term volunteer a bit of course).