The Royal Navy was always drastically cut down in size after the end of a war. The same thing happened after the Seven Years War, for example.
That doesn't mean Britain didn't have the capability to expand the navy rapidly once more. Britain had a huge pool of experienced sailors to draw upon whenever it was necessary. Neither Russia nor America had this capability. It took time and a great deal of money to build ships of the line. Britain's principle of open seas also meant there was no economic gain to be had from supplanting it as guardian of the sea lanes.
The navy had to be reduced in size at the end of the war because it was incredibly expensive (Britain was burdened by large war debts) and so large that it was difficult to find enough sailors to man it effectively: British crews were the smallest of all the major naval powers, which partially explains the early failures in single-ship actions in the War of 1812.
D. Howarth, A Brief History of British Seapower London: Constable & Robinson, 2003), p. 351:
The navy's new work needed far fewer ships and men than the [Napoleonic] war that had ended. In 1815, there were over 700 ships, and 140,000 men: three years later, there were only 130 ships and 19,000 men.
... All those who remained in the navy were volunteers, and that made a basic difference in the sailor's life. The press-gangs came to an end in 1815 and were never used again; so did the system of sending civil offenders to the ships instead of prison.
The Royal Navy spent more time at sea than its potential rivals, giving its men more practical experience and making them more efficient. The post-war navy was a very professional one; more so than the navy that fought and won the war.
"Into the face of the young man who sat on the terrace of the Hotel Magnifique at Cannes there had crept a look of furtive shame, the shifty, hangdog look which announces that an Englishman is about to talk French." - P.G. Wodehouse, The Luck of the Bodkins