The earliest depiction of a bagpipe with bellows is from Praetorius' Syntagma Musicum, 1619. He says it's a French innovation, newly arrived in Germany. Emperor Nero of Rome was said to have worked the "pipes" with his arm, probably a reference to using a bellows to sound a hornpipe, thus a bagpipe. The reasoning behind this was to avoid the disfigurement of the cheeks arising from circular breathing, not to stave off infection. Never heard that one ever.

There is a connection/lineage with the Italian symphonia (only known from various illustrations, a bagpipe sometimes shown with tons of keys), the musette de couer, the Northumbrian pipes, and the Irish pipes. There is a musette de couer said to have been the property of Bonnie Prince Charlie, for instance. Musettes certainly found their way to Scotland and well may have inspired the first builders of the Northumbrian and Irish instruments.

Something that gets overlooked with these pipes is that structurally they are a part of the art music woodwind family, the construction owes far more to contemporary oboes/clarinets/flutes than it does to other forms of bagpipe.