When my in-laws gutted their only bathroom, they try to rig up some kind of shower at home, but ultimately got a three-month membership to the local parks and recreation gym to take their daily showers.

You might just want to make do with washing in the laundry sink. You know, good old fashioned wash cloths. If you get a faucet with a pull-out spray you can wash your hair. You can also do a trial membership at a gym that has showers. :)

Also research DIY camper showers--many of the smaller RVs have water
heaters but no showers, or they have outdoor showers. There are a lot of
ideas on shower curtains and how to dispose of the grey water which is your biggest issue
unless you stand in a rubbermaid tub and shower navy style.

More importantly, get everything absolutely ready and on site before the bathroom is demolished so it's as short a torture period as possible.

Camp shower outside is the way to go, unless it's too cold where you live. Otherwise, PTA Bath in the laundry sink. Trying to make a temp shower inside with a kiddie pool sounds like a neat idea, until you realize how difficult dumping that water out is going to be.

You can by an attachment for your sink faucet that allows you to connect a hose. On the other end you can connect a shower head. I would rig up something like this and enjoy the use of a shower at home while the work is carried out.

I have had two clients in the past do something like this. One used her outdoor deck and the kitchen sink. The other the bathroom sink and the garage.

I think it's worth the effort.

I'm making a base for just such purposes and trying out some different wall panel ideas. Temporary showers - or rather accessible temporary showers are a real demand in any city.

A tankless propane water heater works if you can shower outside. We bought one (an Eccotemp - ~$120 on amazon) when we were without hot water for 39 days after Hurricane Irene. Basement was flooded and so was the hot water heater.

It hooks up to a hose outside, and works great. Every summer after that, we used it for hot water at our outdoor sink. At least outside, we didn't have to worry about drainage. Ours was on the deck with a tarp for a makeshift curtain. We liked outdoor showers so much, we used it until after Halloween. Then it was just a wee bit too cold :-)

Since I'll be doing a down-to-the-studs bathroom gut this summer (hopefully), that's what I'm planning on using. Hope that helps.