In DWDS, fortfahren is described with these two examples:
der Vortragende fuhr trotz des störenden Lärms ruhig in seiner Rede fort
sie fuhren eifrig mit der Arbeit fort
I am wondering if it would be equally correct, or not, to switch the prepositions:
der Vortragende fuhr trotz des störenden Lärms ruhig mit seiner Rede fort
sie fuhren eifrig in der Arbeit fort
I'd say using mit in the context of the speech "Mit der Rede fortfahren" would also work. "Sie fuhren in der Arbeit fort" kinda depends on the kind of work. Like "fortfahren" in that context means something like to proceed with something, so idk picture your finger on the piece of paper an moving it along as you give a speech. So "being within a speech" might make more sense than "being within a job" there are occupations for which that makes more sense and those for which that makes less sense.
Also there might be some funny second meanings which should clear themselves up with the context. Like "mit der Rede fortfahren" could also mean you take the speech (the sheets of paper and literally run with them, like fort (=away) and fahren (=to drive)). Or how die Arbeit could imply both the occupation or the place/context of work. So "an die Arbeit fahren" would just mean to drive to work and so "in der Arbeit fortfahren" could also mean going on an excursion while at work. But again those should clear themselves up due to the context.
