Francis Collinson wrote a fairly good history of bagpipes in 1975 (The Bagpipe: History of a Musical Instrument). More is being considered in the history of the instrument, and obscure references in primary sources are constantly surfacing.

Because the Irish and the Scots especially embraced the instrument so completely, there can be a perception that it emerged as part of Celtic culture from the foggy mists of prehistory. That's really not the case.

Bagpipes are ubiquitous from Iran to Ireland and everywhere in between. They normally use a double reed for the "chanter" the instrument that plays the melody. This image shows chanters for Scotland's Great Highland Bagpipe (top) and for Ireland's uilleann pipes (bottom). It seems extremely likely that this was developed in the Middle East. The first exportation to Europe of this instrument is likely to have occurred during the Roman period (later diffusion during the Crusades introduced instrument such as the oboe and the bassoon). It is not clear if the bagpipe first arrived with or without the bag, although it seems likely that the bag was also a Middle Eastern addition.

The bag allows for the continuous playing of the instrument because of the air reserve. It does not allow for "tonguing" - the technique used for most wind instruments to separate notes, and so bagpipers internationally tend to use elaborate systems of grace notes (quick "chirps") to separate one from the next. A bag also allows for the attachment of a drone - or sometimes more - so that the melody-producing "chanter" can be accompanied by an additional humming "drone" sound. This image shows the complete Great Highland pipes with a long base drone, two tenor drones, a chanter and a blow stick - and of course, the bag connecting them all. This image shows a 3/4 set of uilleann pipes with a chanter (lower left) three drones (alto, tenor, and base), a bellows and the bag. Drones make a single musician capable of delivering a sound something more akin to a band even though there is only one musician playing.

Bagpipes have become integral to most Celtic cultures to the extent that it is an inherent part of the region's folk culture. A Middle Eastern origin of the instrument does not make it any less Celtic - or Iranian for that matter: cultural elements diffuse and are embraced by others. Tradition is not always traditional - unless the adherents believe it is! Perception is everything. This was not really a Romantic-era nationalist thing since many cultures had bagpipes well before that movement, but there can be little question that the push among the Celtic nations to identify themselves with various species of bagpipes was helped along by Romantic notions of "Celticness" and the wish to achieve cultural - if not political - nationalism.

edit: This is an image of reeds from the Great Highland Bagpipe. The reed to the right is the chanter reed, made from two pieces of cane, cut into the proper shape and tied together so the blades vibrate against one another. The reeds to the left are for the drones - two tenors and the larger one for the base drone. They consist of the entire reed, which has been sliced lengthwise to produce an attached "tongue", which vibrates against the body of the reed. Most bagpipes use one or both of these approaches to reeds.