The box joints weren't especially close, so would never hold the drawers together by themselves. To make sure they didn't fall apart, I glued the joints, and then nailed them as well, recessing the nails with a nail punch so they wouldn't show in the final result.
The last drawer came together like this (shown at x15 speed):
It even fit the framework:
Alternating between routing and assembly, I put all four drawers together and - with some judicious filing of edges got them to all fit in the framework quite neatly, which was nice.
Coming up in the next post, fascia and top...