Here are two good threads on the subject:Really? Huh, from what I've read the "twelve apostles" bandoleer was still standard issue (at least among the British) through to the end of the English Civil War. I've even seen some claims that they were still commonly issued during the Glorious Revolution of the 1680s. I always thought paper cartridges didn't really become common until the 1700s.
Ca. 1500 to 1650: Bandeliers for Arquebusiers and Musketeers - Ethnographic Arms & Armour
Mid to Late 16th Century Patrons for Paper Cartridges - Ethnographic Arms & Armour
My general understanding is that a lot of artistic depictions are from later and anachronistically show them in use to the end of the 1600s.