As a player of early instruments myself I can tell you that the instrument you describe sounds to be a rauschpfeife. This is a windcap instrument with the double reed enclosed in a cap. The instrument is very loud and the name literally means "screaming pipe". The shawm is a similar instrument but has an open reed and a more pronounced bell. Both instruments are related to the oboe and, indeed, the shawm is the predecessor of the oboe. Many countries have shawms; the bombarde in Brittany for instance, the piffaro of Spain and the ciaramella of Italy, usually played as accompaniments to the national bagpipes.
My rauschpfeife, incidentally, was made by Gunter Korber of Germany who is now sadly deceased. They are made here in England by Eric Moulder of Leek. The EMS still stocks the instrument I think although decent reeda are hard to come by. I use Kernbach reeds which are excellent but only arrive at EMS sporadically. The instrument is also related in part to the crumhorn, cornamuse and the Glastonbury pipe that you mentioned.
Hope this is of help. Sorry to have rambled on a tad!