One of the files was locked for editing and caused today’s delay in the report's being produced.

…REPORT'S BEING…


'being' is a gerund/verbal noun

compare:

The train was late and caused the delay in ME ARRIVING in time for the concert. : (WRONG)

The 'delay' was not 'in me', but 'in arriving'.. The possessive adjective/determiner preceding it is required:

The train was late and caused the delay in my arriving in time for the concert.