I really hope someone can come up with a more definitive number, for any time period.
From Wikipedia (a dangerous source, I admit) Clayton Knight recruited a total of 8000+ Americans, for RAF and RCAF.
A quote from the RAF Museum web page about Americans in the RAF:
"The Americans who joined the RAF via Charles Sweeney or through the Clayton Knight Committee were the minority. Many more joined alone, usually by travelling to Canada and pretending to be Canadian."
see
http://www.rafmuseum.org.uk/online-e...-1914-1945.cfm
Another quote, from the Nanton Museum web site:
"On 7 December 1941, American aircrew serving in the RCAF totaled 6,129, with just over half, 3,886 under training [BCATP] in Canada. All Americans were given the opportunity to return to the US, and 3,797 requested transfer back to their own national forces. A special train left Washington stopping at every RCAF training base, and 1,759 Americans boarded the train. In total 5,263 Americans completed their service in the RCAF. From 1939-45 a total of 8,864 Americans served in the RCAF, and 704 were killed in training or combat."
http://www.lancastermuseum.ca/s,claytonknight.html