From memory, it goes something like this...
Contemporary reports quoted one of the 1960 works cars as F2-8-60
In 1961 someone (possibly hillclimber Arthur Owen) ran this car
The same year Atkins ran a car numbered F2-8-60 in Europe for Bruce (and JB). Harry Pearce, who built Atkins's cars at the time, insists it was a new car - "Atkins never bought a second-hand Cooper in his life"
The Atkins car Bruce raced in NZ and Australia in early 1962 was, as explained above, another brand new F2-8-60
And to echo what OF said, a friend once asked John Cooper if it was true the factory ran only three lowline F1 cars (T53) in 1960. His answer was, "Well, we only had three chassis numbers...."
At which point the smartest of us give up