The MC and his compatriots are only a step above living on the streets so I doubt they can afford it. That, and they live in Whitechapel, one of the poorest areas in London during the time period it was set in.
I think it would be fairly funny to have the MC tell someone from that timeline that their culture is gross for thinking bathing is bad, and that person just goes, "What are you talking about, we just can't afford it."
How close is the nearest river to where you live? And how clean is that water?
Realistically, if they cared enough they could wipe themselves down with a wet cloth or abrasive, and that would be a fine way to stay clean. It's just that soap is really helpful for killing germs and smells.
They take washtub sponge baths but it's just cold water and maybe some perfume. Soap is pretty hard to get so its unlikely to be used. The nearest river isn't a good idea either because it's filled with tons of dead babies. look it up.
Basically, everything that happened in our history happened. However, magic exists and so do all the worlds pantheons in the form of immortals who just happened to be considered gods by the locals.
The team takes a trip back in time to 480 AD and where they meet Oberon and the MC gets beaten to hell before taking a bath in a healing spring. It's just a hot spring that has magical healing glow moss growing in it.
... I feel that I have seen the premise a bunch of times... so what's this again? MC in medieval London talking to Beowulf and ended up he has to take a nosedive into Nimue's bathwater?
EDIT: Oh. So why would MC complain again? I mean, he can snark and that might be fun.
Nah, MC joins the knights of the round table, stops child abuse with the power of murder, fights in a couple battles, and is taught how to use a sword by an immortal with a nature fetish.
Basically, I made it canon that all Fae are different hybrid children of Oberon and whatever animal or plant he happened to stick his cock in so that I could explain where a good chunk of the world's monsters came from.
Zeus is also real but he was killed during the age of gods and faded into myth just like most of the warlord-like immortals. Also, Zeus knocked people up as a duck so I don't really see the standards.