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?
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.
All I can add to this discussion is that before the use of "follow-up", words and phrases such as revisit, re-examine, re-investigate and other words with the prefix re were used. One can add the prefix "re" to many words but they would have a more specific meaning than "follow-up" which can be used when referring to just about any situation that requires being addressed "again" or "further" at a later time and or date.