I'd like to hear from Hounds who believe the Petersons may have been fellow lodgers at Mrs. Hudson's house. The first is the Glengarry, a military style hat worn tilted to the right, most often seen on overseas duty. Glengarries are often seen on bandsmen and others required to wear fancy-dress. The other kind of bonnet is the Balmoral. It's a beret-style hat worn forward on the head, with the top pulled gently to the right.
Balmorals are both diced and undiced.
The ribbons at the back must be tied in a neat bow. Alas, neither Glengarries nor Balmorals are quite right for this story. The Oxford edition's choice of a Tam O'Shanter for Henry Baker's bonnet, though not technically correct, is the ideal alternative. Tams are round, flat, soft woolen caps, sometimes knit in one's tartan colors, often with a large toorie on top. One source likens a Tam to wearing a tea cosy on one's head, and that's not a bad comparison.