My grandfather used to make toys similar to these for a charity called Toys for Tots. To the best of my knowledge he built everything with a couple cordless drills, a 14" bandsaw and a multi-function sander like this one, and copious amounts of hot glue.
I personally would add to this a spindle sander and at least a skeleton set of quality hand tools(chisels, rasps and riffles, block plane etc.) but I'm more into the fit and finish of things than Grandad was. Basically you need to be able to:
- dimension your stock
- fit and shape your parts
- assemble the parts
Given that the parts you'll be working with are generally fairly small, a bandsaw is a great choice because it's much safer than a table saw for short pieces and it can cut . And the sander is limited only by your imagination in terms of the variety of its uses. You can even use it as a lathe in a pinch (run a hanger bolt into the work piece, chuck the end of the bolt up in your drill, spin the piece up against the sander and viola)