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According to the question Are toilets renovated at Hogwarts? and its answer,
[Before toilets were added] they simply relieved themselves wherever they stood, and vanished the evidence.
However, new students, and particularly the Muggle-born ones, start at Hogwarts without much knowledge about magic. How did they manage to relieve themselves and clean it without even knowing how to cast a spell?
I assume they used chamber pots.
Chamber pots are definitely a thing in the Harry Potter world -- Dumbledore has used them at least once while in Hogwarts (emphasis added):
“Oh I would never dream of assuming I know all Hogwarts’ secrets, Igor,” said Dumbledore amicably. “Only this morning, for instance, I took a wrong turning on the way to the bathroom and found myself in a beautifully proportioned room I have never seen before, containing a really rather magnificent collection of chamber pots. When I went back to investigate more closely, I discovered that the room had vanished. But I must keep an eye out for it. Possibly it is only accessible at five-thirty in the morning. Or it may only appear at the quarter moon — or when the seeker has an exceptionally full bladder.”
Harry Potter and the Goblet of Fire, chapter 23: "The Yule Ball".
They relieved themselves the muggle way.
It is pretty common for witches and wizards to be practiced in some of the more mundane muggle arts, and often possess many of the tools requisite for the crafts. For example, why would a magical family own cleaning instruments like sponges or dusters if they can just magic away messes? Yet we know that Mrs. Weasley makes her children often clean with these tools, sometimes as punishment, though probably often to get help from the younger ones who cannot use magic yet.
Children who do not yet have the right or ability to use magic at home would certainly use a chamber pot. It stands to reason that there would also be such accommodations at Hogwarts, as well as most other magical institutions.
This piece of info is actually news to me. Even if Hogwarts didn't have the bathrooms and equipment that it did later, I am sure that even when it was founded, it had something that all wizards and witches and muggles need - which includes bathroom facilities.
What was in the update regarding relieving themselves on the spot and vanishing the evidence would have been for some mischievous kids who didn't want to go some specific place or do some work when they had wands and magic. Of course, I agree that it would have made stuff miserable for the house elves.
Thinking further into the same - they could have used the room of requirement or like Dumbledore once said that he found a room only when he needed a bathroom. Maybe there are hidden bathrooms that were commonly used then. After creation of proper plumbing and bathrooms, people forgot about them. Since those rooms only appeared when required and no one required them anyone, there wasn't any issue.
