Injuries in the 3 batter rule

Unfortunately, MLB has not published the 2020 rulebook yet; the newest they have available is the 2019 rulebook, which was before the rule change took place.

What has been reported, verbatim on many sites including mlb.com's article on the subject, is:

All pitchers -- both starters and relievers -- now have to face at least three batters (or pitch until the inning is over) before they come out of a game. The only exception is an injury or illness that prevents the pitcher from being able to finish his three batters.

The main effect of this rule will be on specialist relievers, who are often used for only one batter to give their team a favorable matchup -- for example, a left-handed pitcher who faces only left-handed hitters, whom he is most likely to get out (often nicknamed a “LOOGY,” for “left-handed one-out guy”). There will be no more of that in 2020. The rule could also impact teams that use openers -- typical relievers who start a game to match up specifically against the top of the opposing order. Now, an opener would have to face at least three batters even if he has a bad matchup in that run.

From that wording, it seems likely that the new replacement pitcher would have a new 3 batter minimum; but until they publish the offiical rulebook update it's impossible to tell.

I did find something that might be the 2020 official rules - they seem to be, anyway - at Baseball Rules Academy. That just confirms the language above (in rule 5.10g):

(g) If the pitcher is replaced, the substitute pitcher shall pitch to the batter then at bat, or any substitute batter, until such batter is put out or reaches first base, or until the offensive team is put out, unless the substitute pitcher sustains injury or illness which, in the umpire-in-chief’s judgment, incapacitates him for further play as a pitcher.

The starting pitcher or any substitute pitcher is required to pitch to a minimum of three consecutive batters, including the batter then at bat (or any substitute batter), until such batters are put out or reach first base, or until the offensive team is put out, unless the starting pitcher or substitute pitcher sustains injury or illness which, in the umpire-in-chief’s judgment, incapacitates him from further play as a pitcher.

It does not specify any special rules for the injury replacement, so unless there are umpire notes (again, not available at the moment from what I can tell) clarifying this, I would assume the injury replacement must pitch to three batters.