Could the filming of the 747 in "Awakenings" have taken place during the 747 flight testing?
Nope. It wasn't filmed during testing.

When Dr. Sayer takes Leonard out into the city for the first time, they see (among other things) an aircraft landing - a Boeing 747. The movie takes place in 1969 and the first revenue flight of the Boeing 747 was January 21st, 1970. Additionally, the 747 is in El Al livery that was used around 1990 (IMDb).


(airliners.net) Photo date: October 27, 1991.
Could the 747 have been filmed while landing at JFK in the summer of 1969?
It's plausible, because during the flight testing the 747 flew to Paris and it could have very well used JFK (runways were upgraded in the late 60s):

[Boeing took] a test aircraft to the 28th Paris Air Show in mid-1969, where it was displayed to the public for the first time (Wikipedia).

and

[JFK] was designed for aircraft up to 300,000-pound (140,000 kg) gross weight and had to be modified in the late 1960s to accommodate Boeing 747s (Wikipedia).