Reddit mentions of Intermatic FD60MWC 60-Minute Spring-Wound In-Wall Countdown Timer Switch for Auto-Off control of Fans and Lights, White

something like this should do the trick

Why not just use the tumble dryer - then the clothes come out all nice and warm!

But the best way might be to get a timer swtich, a space heater and build a large-ish sized box that can fit a complete load of laundry on --- maybe 3 ft long, 3 feet deep, 6 feet tall --- with stainless steel pipe or plastic pipe that can be used to lay the clothes over.

Put the space heater in the box, hook it up to the timer switch however you'd like, connect electrical power to the big box however you'd like (by creating a cord with a plug that you plug into the house power or by running house power directly to it) and then test it.

The space heater should have some temperature safety setting to make sure it doesn't get too hot. The switch ensures you don't forget and leave the heater on when you leave the house. The box keeps the warm air in the box to help save power.

With a temperature safety on the space heater you can set the heater to a specific temperature to avoid shrinking wool clothes. You'd have to do some research into the proper temperature.

Based on what you described, this is what I'd do, in general terms.

EDIT: Here's a Picture of what I was thinking of. A small exhaust hole at the top would probably help equalize temperature so the top of the box didn't get substantially warmer than the bottom (which you probably happen without a small exhaust hole at the top.

EDIT: Suck air from basement and try to exhaust it out of the house. Thanks to /u/iproblywontpostanywy for pointing out you'd end up with a lot of humidity in the box.

For bathroom fans, as much as I love tech, I use old school, because it works:
Intermatic FD60MWC 60-Minute Spring Loaded Wall Timer, White https://www.amazon.com/dp/B00002NAPD/ref=cm_sw_r_cp_tai_vN7OybPXX2KN0

Whats the timer for? If you dont have to use it for any fancy control and it is just a bath fan or something you can use a mechanical Timer

Air out that thing.

Open all Windows and doors at least once a day for 20+ minutes to reduce the humidity.

Get a induction cooktop, instead of the Gas burner - burning with Gas produces a huge amount of water vapor.

Leave the cabinets cracked open.

Every time you use the bathroom - use the fan

- get a fan timer: https://www.amazon.com/Intermatic-FD60MWC-60-Minute-Spring-Wound-Countdown/dp/B00002NAPD

So anytime you use the bathroom - put it on 10-15 minutes.

I have been happy with https://www.amazon.com/Intermatic-FD60MAC-60-Minute-Spring-Loaded-Lights/dp/B00002NAPD/ - I tried an fancier electronic one that didn't work, now have two of these in two different bathrooms.

It sounds like you want a timer not a thermostat

Intermatic FD60MWC 60-Minute Spring-Wound In-Wall Countdown Timer Switch for Auto-Off control of Fans and Lights, White https://www.amazon.com/dp/B00002NAPD/ref=cm_sw_r_cp_apa_i_5r-kDbWPVAFP5

Can have one for cooling and one for heat, would just take a bit of wiring

If you want a smart one you could use some zigbee relays and IFTTT to make it do what you want.

Because the nest thermostat is not going to do what you want

I think I'd rather have a spring-loaded model like what is currently there, apparently they still sell them: http://www.amazon.com/Intermatic-FD60MWC-60-Minute-Spring-Loaded/dp/B00002NAPD/ref=sr_1_5?ie=UTF8&qid=1420672697&sr=8-5&keywords=wall+timer+switch