From both the picture and the drawing it appears to me the pickups DO have bobbins. In the picture they look like clear plastic and in the section drawing they look like ][
The top photo shows the coil wound directly around the magnet. This is fairy common for some older pickups to not use bobbins, like Burns Tri-Sonics. That was a prototype pickup made for Les Paul for one of his personal guitars. But I think it was established here that the Triumph pickups did have some kind of bobbin.
The bottom illustration is for the LP Recording pickup, and that did use bobbins. They look like this:
That's a Triumph cover, but Recording bobbins.