Thanks guys - good thoughts.
Today I ran some more controlled tests, using the simple test flight plan, three of my AI a/c (not including the newest one) and two of my airfields, including the one where I first identified the problem. The results were interesting.
All three a/c taxied, took off, flew the pattern, landed and taxied back to the parking spot at both airfields as expected.
One a/c communicated with ATC correctly throughout all phases of flight in the air and on the ground at both airfields.
The other two a/c exhibited the problem described above at both airfields i.e. they requested/received taxi clearance from Ground, requested/received takeoff clearance from Tower and took off. Thereafter all ATC communication ceased.
So although there may be an airport factor, this would seem to be an aircraft problem. So I started comparing the 'good' a/c and one of the failing a/c, which are fortunately quite similar. Following Guenther's tip above, I checked the Radios section in the aircraft.cfg files, and they were different, so I replaced that section in the failing a/c cfg file with the same section from the 'good' a/c. This did not fix the problem.
So I got a bit more heavy handed. Since the two a/c are similar, I was able to swap the .air files and the aircraft.cfg files. And I found that if I swapped BOTH, then the problem was fixed. However, if I swapped only the .air file or only the .cfg file, the problem was still there.
So I'm guessing there's something in the failing a/c .cfg file that is overriding something else in the .air file and somehow causing this problem. Now to go through the two .cfg files to see what I can spot.