Washing in the absence of soap
Soap is cheap right now. Even with inflation, it's easy to buy solid soap or concentrated soap and then dilute it for the job. Thinking like a Pioneer, imagine not having soap.
While it's relatively easy to make it since it only requires two main components: fat and lye, it will require making some special tools if you don't have them at all. That means collecting the lumber, construction times, several bowls, and time for it to prepare and cure. If you think about it, the lye is very easy to acquire from wood ash, but the issue is the fat. While oil, or tallow (Crisco or Lard or animal fat trimmings from the butcher) are cheap now, finding an animal of sufficient size to harvest the fat is not at all easy.
Imagining that it's possible for STHF, then you've stocked up. Still you'll run out. That means anticipating that, you'll properly dilute your existing soap sufficient to clean, but not overdoing it. If it's too concentrated, it takes far more water to rinse it away. A tiny amount is really needed to remove debris and dirt and natural oils. You've got to reteach your family how to wash with limited resources.
You might also be working with a limited water supply, which means be very careful about washing and timing it based on water availability. I've discussed hygiene and it's importance. Now here's a very basic way of staying pretty clean when you've run out of soap.
There are three main things you've got to attend to: cleaning yourself, laundry, and for cooking. It is best to have separate cleaning bowls for all three. That's tough if you're bugging out. As I've detailed, when bugging out, it's a major undertaking to gather enough water and get everyone clean.
Bathing and Washing Up
Without oil or tallow it's hard to get really clean, but you can just about always make lye from a tiny bit of wood ash added to your wash basin. A second rinse basin will have some hot water added to it from a central pot that stays heating on low on the stove. For folks who are modest, it's pretty easy to do this within a tent or to string up a tarp. If you can fashion up a place with runoff on a slightly pitched hill and some stones to help it to drain, then another person can use a rinse bucket to pour on whoever has some of the mild caustic lye on them. Mostly, you'll be using the scrubbing action of a towel to wipe away cellular debris and dirt with the toil, and using the lye as a disinfectant to knock down the Strep or Staph bacteria that will accumulate on your skin. Lye feels soapy, but can burn the skin, so you make a really weak solution and wipe away and rinse thoroughly. Be careful around your eyes. That's why soap burns them from the lye.
You'll be washing your hair far less. You'll be cleaning off simply with a wet portion of the towel and drying with the other end. When done, you'll squeeze the towel in the lye water and hanging it up to dry. This means timing it when it's cold out and based upon the availability of fire to heat up the water too. You must always clean off hands and feet and face each day. Keeping your feet clean and free of things like cracking, dirt, long nails, and fungus is crucial. You'll be walking WAY MORE than you've ever done in your life. It's far easier to file your nails done every other day in seconds then trying to clip them without a clipper. There are lots of sandstone pieces you can find to do that in the absence of a clipper.
You'll need several pans, several bowls, and a cutting board when cooking. You'll make a bowl of the lye water and run your hands through that and then quickly rinse them as you touch things and keep on cooking. A dedicated kitchen towel will allow you dry off your hands. You'll be minimizing dirtying up dishes to save time cleaning. Making one pot meals and a starch like hard tack or a corn bread will mean less dishes too.
When handling raw meat from harvesting an animal, it's very easy to taint other containers with food in them while preparing the meal. This takes practice. If you don't exercise care, then you can easily introduce Salmonella or Shigella into the stew. People don't normally live out of tens for very long, otherwise they encounter this. Using this system will really cut down on intestinal flu.
Pots get really black from cooking with wood. The best fires are coal fires. Mostly it's the inefficient flame fires that produce lamp black. Rubbing the pans with ash and water and a little sand before will make cleaning the outsides way easier. In addition, the lamp black will get on everything, so using some care about repacking your cooking gear will help. A net like a netted laundry bag makes a great way to carry them. This allows them to dry better and also removes the direct contact with other gear.
Locating some sand, and tossing it into the boiling water, will give you a way of scouring your pans. Try not burning any food as it will make cleaning the interior miserable. Obviously some water poured inside and then boiling away the debris makes this easier too. If you've added a lot of starch from tubers like cattails, this can burn in your pans, so be careful.
Likewise, it's easy to find a tiny pine tree branch as a ready made scrubber. It's also an antiseptic (from a little of the resin), so it's doing double duty.
You'll be wearing clothing more often. You'll be making a little stronger lye solution to wash. Take care of your hands as you'll be doing a lot of hand wringing, unless you modify a mop wringer for the task. You can rig up a washboard pretty easily, but try not to do this too much as you're breaking down clothing fibers. If you've got minimal clothing, you're managing the wash based on who can borrow from another family member. It takes time to dry out clothes or boots, so care must taken. Wet clothes or boots chaff terribly. It meant you did a lousy job timing the wash.
A little cattail pollen or dried tuber will produce some starch. This is great for chaffing. You'll be no doubt drinking a lot of pine needle tea for Vitamin C. It also produces tannin, so that will help cut down on bacteria in your mouth and breath. Dabbing a more concentrated tea under the arms cuts down on bacteria which is what causes sweat to smell. That's important for hunting as animals will smell you or hear you before they see you.
A little saved tallow will also be helpful for chaffing and for treating your hands. Oils collect in your hair and along your nose, so wipe that into your hands from all the lye being used. This is very important in winter as the humidity declines. Obviously rubbing your hands though hair and on your face will prevent cracking. In times other than winter, scrubbing the skin a little firmer removed the dead skin and giving a health glow. Use caution doing that in the wild as you are exposed to higher bacterial counts and you may get more pimples that way.
You can brush with a little ash or charcoal you ground up. It sound dirty but it actually works and cuts down on bad breath. Mint is found lots of places, and is easy to identify and is prolific, so when you find it, you'll gather it for treating stomach ache, a welcome and calming tea, cleaning the sinuses, making the temporary shelter smell better, etc. It also is a great descenting herb for hunters as mint is s common.
Later after not having soap for so long, you'll see why making soap is so important. If you're lucky enough to have some tallow, then you can make a small batch and let it cure as you travel. Mostly since food may be very limited, you'll be using that fat for essential calories. However at times it comes in all at once from a kill, so that's an opportune time to make tallow lamps, tallow laced cattail candles, or soap. Remember that pine resin is plentiful, and that can be used to make up homemade pine oil cleaner, but dilute it. It's very medicinal too.