The earliest paper cartridges I've seen date from the 16th century; one end was plugged with the bullet. They seem to have been rare, and used with hunting weaponry. In the 17th century you start seeing paper cartridges (perhaps more properly "chargers"*) being used in military units, but it's not until the beginning of the 18th century that they become standard.
One reason paper cartridges may have taken a long time to become common, is the general lack of standardization in general. It took sometime before weapons were being made standard enough that a central supply system could provide preloaded ammo. Since the 16th century, large groups of standardized weapons might be ordered, but I think the usual requirement is that they be of the same caliber -- not that they match some national standard caliber.
*I would refer to the wooden tubes with a preloaded amount of powder in them as "chargers" and not a cartridge. In my mind a cartridge combines powder and ball, whereas a charger contains only powder. I don't know if that's a common distinction.