Living alone in an older house without such great insulation, and not liking to spend money that much, I keep my thermostat real low at night. All I do in the morning is jump out of my toasty bed, take a shower, get dressed, and go to work. It seems like a huge waste to heat the place up just for that. And even for most of the winter, it doesn’t get all that cold at night to drop it too far really.
But in these polar vortex situations, it really makes me hate the post shower situation. The instant after the wonderful hot shower turns off, the encoldening begins. Snaking my arm around the curtain to grab the towel while maintaining as much of a seal as possible to prevent cold air from hitting wet skin. Then frantically toweling off as the cocoon of warm air dissipates rapidly over and through the curtain, before my scrotum pulls up into my kidneys.
How I dream about a way to keep that great warm, steamy air in place for a leisurely post shower drying off and stretching.
A simple electric space heater works fine in a small enclosed space like a bathroom.
Attach a timer, and it’ll turn itself on a 15 minutes before you wake up.
Problem solved.
How big is your bathroom? Mine’s not much more than about three times the size of the tub.
You could also look at a plug in towel heater like this one, at least your towel will be toasty and will dry more quickly once you’ve used it which keeps it from getting musty smelling.
One other possibility, and much more expensive is to replace the flooring and add electric heating but that is a lot of work if you DIY.
Make sure you use a timer that can cope with the current/power requirements of the space heater.
We use a space heater in our bathroom as well. Like the OP I towel off in the shower as much as possible before stepping out into the room - and the space heater is right there, so I am stepping into a warm, gentle breeze.
Assuming 13 cents per kilowatt-hour, A 1500-watt space heater operating for half an hour (15’ before shower + 15’ during) should increase your electricity bill by about ten cents. Shower every morning? Your monthly electricity bill would increase by $3. That’s probably a better deal than trying to warm your whole house up every morning.
I have a programmable thermostat. It has four temperature points, wake, day, eve, & sleep. I keep it cool during the day & overnight, but warmer for morning routine & in the evening. The morning bump up isn’t that long. It’s only a few seconds to change the program to bump it up higher in the morning during the vortexes.
Ranging from about $30 to $100 (the cheaper one have less programmable days, meaning it might be one program for all seven days or a 5/2 where the more expensive ones will allow you to set it for each of the seven days.) it’s much cheaper than enclosing the shower to the ceiling.
My wife showers first, pre-warming the bathroom. And she throws my towel on the radiator so it’s toasty warm when I get out. 
Well that’s what you need, then - Telemark’s wife!
I would get a radiant space heater (actually I have exactly that already), the type that glows orange and usually does not have a fan. Just snap it ‘on’ and it is instant heat, turn off as you are ready to leave the bathroom.
Also the towel over the shower curtain rod, so no need to open it up.
The programmable thermostat with the morning burst of heat is nice too.
I use all 3.
You may also consider using something to partly seal off the top open section, perhaps some plastic.
They make dedicated bathroom heaters, like this.
Another option is (if the bathroom has a fan vent) to replace/augment the bathroom fan with a heat lamp like you see in many hotels.
Knowing what it’s like to take a hot shower in a cold house, you either have to seal off the shower quite well or not try at all. Due to the circulation of hot air from the shower and the cold air from the room, if you close off the top, the air will do everything in it’s power to go through the curtain. In which case it will immediately suction onto your body like a cold, awful, clingy blanket and not come back off. So, due to the low ceiling, showers in that house had to have the curtain cracked open at least a foot to allow the air to circulate without blowing the curtain in all over me.
Then again, I always used a single lightweight shower curtain, not that heavy duty plastic stuff with an extra fabric part on the outside. The weight of those might make it ok to close off entirely.
Really though the space heaters are a better bet.
get a blanket storage bag, with a zip lock, place towel in it and hang on shower curtain rod.
You could try a double sower curtain, on the same rod, one inside the tub one outside.
Also for hanging the towel I would recommend making a hook out of a hangar and hook it over the rod