What word was used before 'follow-up' emerged in 1905?
Did a single noun precede the hyphenated compound noun 'follow-up' (whose emergence is in 1905 is asserted by Etymonline)?
Did a single verb precede the phrasal verb 'follow up'?
If the answer to 1 or 2 is 'no', then what nouns or verbs (respectively) had been used?
None of these synonyms proposed by ODO or Thesarus.com synonymise sufficiently.
A word that would have been used in the same manner as the noun "follow-up" is "prosecution." The Oxford English Dictionary provides several definitions that support this. The first listed states, "The following up, continuation, or pursuit of any action, scheme, or purpose with a view to its accomplishment or completion," and if this does not convey precisely the meaning you are associating with "follow-up," the OED offers a number of others, as well as quotes that demonstrate the usage of "prosecution" in a synonymous manner, dating from the 16th century, but I believe this one best captures the meaning: "This Chapter is a prosecution of the latter end of the foregoing." The online version of Merriam-Webster Dictionary offers only court-related definitions of "prosecution," but dictionary.com, with its second definition (the first relating to court proceedings), "the following up of something undertaken or begun, usually to its completion," supports the OED definition.