Not quite the period you're looking for but the systems I was working on in the 1980's for the Admiralty still used punch cards (for the results of compliation of code on a VAX/VMS system that could then be ported to the transistor and valve computers the Navy were still using back then, which were resilient to EMP).
I also did my O level in computing via a golfball teletype machine and an ICL2904 mainframe. The results of which were sent back to us as a stack of punch cards.
The reason punch cards are faster than a manual system is that they allow cross-tabulation of results very quickly and without the man-power necessary in a manual system. They also allow you change the program code easily by substituting cards.
Sorry if this a bit vague, but I'm a bit rushed for time atm.....