This is probably regional as I don’t use either word that way in Québec.
I never use serviette in English, and in French it’s a term used for a towel or something to absorb liquid with; serviette de bain, serviette de plage, serviettes en papier (paper towels). Though we would call a maxi pad a serviette sanitaire.
Napkin, in English, is a napkin as you described, to wipe hands and mouths with.
Comedian Trevor Noah, who hails from South Africa, has a bit about this. He says a guy is telling him about how good this food truck is; it’s so good that you need a bunch of napkins.
Now, Trevor had heard of the reputation that Mexican food had with intestinal distress, so he was horrified that this guy was lovingly talking about how you’d need a napkin to wipe up all the mess that would be made after you ate.
“Bath tissue” got coined because people were weird about saying “toilet” in public. My understanding is that, for much the same reason, Americans say “I need to go to the bathroom,” when what they are really going to do is go to the toilet.