As far as I understand the problem with bathing (I don't KNOW for sure, but it seems logic to me), the problem was that bathing ban came after roman empire public baths. They were claimed to spread disease, which was absolutely true! If a number of people swim in WARM, non-chlorinated water, it would only cause all bacteria and viruses those people carry on their skin bloom and spread.
Mechanism of disease spreading was completely unclear. Bacteria and viruses were unknown, and even unimaginable.
Also to have really clean water for bath you need first to boil it and then let it cool down to usable temperature. Mixing it with unboiled water would most likely cause all bacteria multiply (and you won't like to have even a tiny scratch on your skin in such situation, as the wound would immediately get infected).
Fats, required for soap production were rare and expensive. People had to hunt for whales for their fat, as they had to have something for illumination. Beeswax for candles wasn’t enough in that meaning.
The last, and most important, boiling water and making soap requires a lot of heat, and that amounts of fuel could only be achieved with mass coal mining . And even after the coal is mined, then it needs to be delivered, and with horse power only, it is a great problem. Only Britain had so much easy accessible coal, short land distances and sea all around.
Of course, all of this is my personal opinion, based more on logic (as far as it is understandable to me) rather than solid historical facts.
I apologize for any grammar or spelling mistakes, English isn't my first language.