1. Made With Glass: Large glass timers of this nature aren't exactly expensive these days. You can find them on the shelves of TJ Maxx, Homegoods, Crate and Barrel, and all over the interwebs. Purchase one and then construct the rest around it according to its dimensions.
2. Made With Plastic: Not comfortable using glass around kids? Yeah me too. Instead, try using clear soda bottles. Cut the bottoms off two different bottles, remove the lids and secure in the middle with a large wooden bead that has a hole drilled through it. Secure it all with your favorite adhesive (I'd turn to E6000) and continue with the rest of the stool assembly.
Even if you're not a hardcore woodworker, you can still knock this project out. Don't assume it has to be round, buying precut squares at the hardware store is easy and having them cut for you from your wood of choice is just as simple. Pick up spindles in the decking section or from Craigslist.
The only word of caution is to make sure that the outer supports or spindles support the full weight of the piece. If there's no weight on the timer itself, then there's little to no fear of it breaking no matter what it's made from.
If you're not into the DIY version you can pick up a Time Out Timer Stool for $69 from Wisteria.