Thanks go out to my brother who came out and installed the switch below:
He brought along a new rotary timer switch which will go to good use in our other bathroom.
We already had the switch above and primary user of this bathroom prefers the bush button style.
All good. Feelings of gratitude.
December 2nd update-
This video looks like it may be the way to go here:
Original blog entry:
All is working except bath fan timer. Fan is good. Timer is faulty.
Spent some time on the web and have not made progress.
Not looking to do a home remodel.
Does anyone know:
Is the old part available?
Will a new style timer fit in this old switch plate?
Anyone who can provide info will be much appreciated.

If you’re not making progress fixing your old fan, just install a new one. I installed this one myself: Panasonic WhisperFit DC LED Bathroom Exhaust Fan. All the white wires go together as one, black to black, and the third wire is the ground, which connects to the ground wire.
I managed to get it done in about 30 minutes without any “remodeling” or needing some overpriced “fan installation specialist” from Craigslist. My old fan broke, so I tossed it, bought this one for around $200 (including taxes), and solved the problem. Huge thanks to YouTube for the installation guide-you can absolutely do it yourself! Link: https://www.homedepot.com/p/Panasonic-WhisperFit-DC-LED-Pick-A-Flow-50-80-110-CFM-ENERGY-STAR-Quiet-Ceiling-Bathroom-Exhaust-Fan-Flex-Z-Fast-Install-Bracket-FV-0511VFL1/317716369
Thank you for your comment.
Primary goal is just to replace the timer. The fan still works as new.