I have vivid memories of late nights copying out big band parts by hand to get them ready for an upcoming rehearsal. Making a mistake in the part involved either trying to scrape the ink off the page while leaving it still legible, literally pasting a cutout over an entire staff to cover the mistake, or just starting that whole page over again. Once I got access to music notation software I gave up writing out parts by hand (and the musicians were very happy to not have to read my mess too).
Before music software was prevalent, individual composers and arrangers had few options beyond writing out everything by hand. Publishers developed a variety of techniques over the centuries, ranging from wood blocks to moveable type printing presses to this unusual typewriter, a Keaton Music Typewriter I happened to notice for sale here. (I love what appears to be a coffee stain on the left side of the paper over the treble clef.)
I imagine that putting together parts this way would be tortuously slow, at least until you did enough of it to get faster. Even then, I suspect it would be faster to write out the parts by hand, but it wouldn’t be as easy to read. Correcting mistakes would be just the same, unfortunately. I’ll stick with my computer notation for now.
For more on how music has been printed in the past, visit the online museum, The History of Music Printing.