American bases were established in Bermuda in anticipation of American involvment in WWII. Originally Fort Bell and then renamed Kindley, it was operated by the USAAF during the war. The USAF was organized in 1948 and the base (later as a U.S. Navy NAS) was operated until 1995, when the facilities reverted to civilian usage. See below for a history of U.S. military bases in Bermuda.
I do not know the exact dates the American school was in operation.
Web sites:
Kindley AFB High School Web Page (alumni)
Kindley Air Force Base (Wikipedia)
American Military Bases in Bermuda 1941 - 1995