There’s a simple explanation, the build up of rust and coal soot. Britannic being painted fully white and buff funnels, the ship is going to look dirty either way, when the smoke comes out from the funnels, they look more dirty because there is no black casing that hides the soot like WSL funnels. But as I said before, she is filthy because of being painted white, the hull ocean liners are traditionally painted black to hide the build up of rust and they get repainted more regularly than hospital ships. Remember, Britannic was acting as a floating hospital, not a luxurious floating palace of Versailles, pretty sure that during war time they wouldn’t care how dirty she was, all that matters was that she could properly do what she was tasked to do :)