I think there was an engine change from Coventry Climax to a Standard Special in July 1939. I am not sure how many Standard Special engined cars were produced before WW II.
The question about post war Morgans is very difficult since the wars lasted much longer than we normally think.
Formally World War I ended in 1958 and World War II in 1990:
Peace Treaties.Harald
I'm no expert but on all the information I can find, production figures for the 4-4(1936-1939) and 4/4 (1946-50) models were: a total of 1428 ( 1307 standard cars, 113 Rolling chassis for specials, 8 le Mans and TT Replicas)
The original Coventry Climax engines ( The Triumph 'Gloria' engine) were of 1122cc but from March 1939 a smaller, 1098 cc engine was offered as an option. The Standard Special engine was offered as an option from May 1939
As John Worrall says, there were a number of external differences- such as some cars running on 16" wheels and some on 17" butI think with anything short of a very close inspection of an individual car it would be difficult to identify general differences
One general point which CAN be used to tell Climax and Standard Special cars apart is that on the Climax cars the tail pipe exits on the left hand ( near) side but on the S/Special cars it is on the right ( offside)
HTH
g