These weren’t woodcuts — the prints were made from photographically manufactured plates made from drawn originals, essentially "analog scanning"

This technology emerges in the 1890s, which is why you see a big increase in daily cartoon content at that time; there were engraved illustrations earlier, but they were much more cumbersome as a technology.

So the cartoonists drew with pen and ink, the artwork was photographed, the photographic plates used to prepare the printing plates; the masters could be sent to other newspapers to be printed locally.