It's cold at the moment and I can't spare much for heating the house.
Are there any cost-effective hacks to hang-dry wet clothes faster (and preferably limit mould-inducing moisture in the air)?
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I find that drying is more about the amount of air being moved around more than the temperature, sure a hot tumble dryer or hot radiators do the job the best but it doesn't mean its the only way.
So while it isn't perfect (as in near zero cost) a fan in front of your clothes will assist with the drying process, Another option is to keep your clothes hanging high in the house where it'll be a little warmer than ground level, before the creation of tumble driers a lot of houses had rails on a pully system that was used to dry clothes on.
If you can afford it, a dehumidifier wins hands-down. 1) your clothes dry faster, 2) moisture is REMOVED from the air, not added, 3) all the energy used (possibly more) is returned to your home as heat.
Removing excess water should be done first. Lay out a bath towel with the garmet spread out on the towel, roll the towel up, then stomp on it. That will remove almost all of the water. If most of the water is removed, clothes dry pretty fast anywhere they are hung. Hanging near an air vent helps, too.
To keep the water from going into the air and causing problems, you need to get it out first.
The best option would be to iron your clothes. Not only will it dry the clothes, but it will keep them wrinkle free too. Also, the iron will use a lot less electricity than what would be needed to heat the whole house.
The second best option would be to wring dry the clothes. You can do some by hand, but usually I find that it's hard to get enough pressure on the fabric to press the water out. If you have two rolling pins in your kitchen, you could use rubber bands to keep them together, then pull your clothes through the rolling pins. Otherwise, lay your clothes out on a table and roll the water out of them with the one rolling pin. That should let you apply enough pressure to get the water out.
As a last resort, you could use a hair dryer and dry your clothes that way. It's less electricity than heating the house, but I doubt by much when you're done drying the clothes.