Do you have a model number and MFG name? That would be helpful.
Make sure the Breaker is OFF or the Fuse is OUT before you work on it. Do the following at your own risk.
From my Experience, about forty year, the white wire should go to the neutral wire, another white wire. Which is not one of the original wires. You may need to pull one in or fish it out of the box. The Black wire goes to the original black wire, which should be Line or hot wire. The red wire should go to the Whitish wire, this should be your load or light. Make sure you use the right size wire nuts and use electrical tape to ensure the wire nuts will not come off.
Your timer switch needs power to run it. It uses the black and white wires for power. It has a relay inside it that also uses the black. The relay connects and disconnect the red wire from the power, the black wire.
I hope this Helps!