You also presumably have a cold water tank in the loft. This is more likely to be dirty and stagnant. It might supply the cold bath tap. To check, turn off the main stopcock and open the bath tap. If it runs, it will probably drain out the tank in about 15 minutes. Give it a clean and rinse, refill and drain again.
If the tank does not have a loose-fitting plastic lid, it is particularly likely to be dirty. If it does not have a full insulating jacket, it is particularly likely to be stagnant.
So fit these accessories if they are missing.