Not gonna lie: when you said "Cardboard clock..."
I think I'd read through the first few sentences when I realized that you weren't talking about these old paper clocks. Still...the best part is that you actually did bring a working clock to school. Yours worked. Theirs didn't.
The shame about it is that it could have been a teaching moment. I'm visualizing a better teacher getting out a flashlight or something and showing the class how the extra step resulted in a working clock. Good teachers can see past their lesson plans; I had a couple of elementary school teachers who would have headed to the blackboard and run with it.
If you're really looking to make your own clock from scratch, you might want to check out this book. It shows you how to make your own clock out of paper. It's pretty tedious because for instance, to make a gear, you have to cut the two sides of the gear out and then wrap a third thin strip of paper around each tooth. The book has you cut out the parts and assemble them, but you could trace them and cut them out of some pretty thin wood.
As far as the self-immolating feature, this pendulum clock is gravity fed, so you could just make a mechanism that would be triggered by the weight. Just lengthen the string so that it gives you 7 days of non-stop clockage.
Hey, if the string is a fuse, you could have the weight slowly lower itself into a flame :)
What about a cut out book that you can make a working clock from the paper?
https://www.amazon.com/Make-Your-Working-Paper-Clock/dp/0060910666
Not a big fan of amazon but it's something that you could do with him and it would work.
This is the specific book he had. The pages of the book are the materials, if that helps.
I bought a copy, I'm excited to put it together. As an aside, have you ever considered making a clock, like in this book?. It would probably turn out pretty cool looking
You can make a clock entirely out of paper:
http://www.amazon.com/Make-Your-Working-Paper-Clock/dp/0060910666