It might not have been the first modern bath, but according to the Country Life archives- It was the prettiest by far in 1920. The address 90 Gower Street, belonged to Lady Diana Cooper. Her mother, the Duchess of Rutland was the artist of this Chinese paper used in the bath above. The Lady remembered :
'We took a tracing of a Chinese paper at Belvoir and together on ladders we painted the white trees and birds and cages and butterflies on a pale green ground. It had a marble perspectived balustrade and ,as at Belvoir, a marbled dado. The bath was hidden in a lidded coffer marbled to match. There was a large sofa, a pretty fireplace and gilded looking-glasses...All the floors were carpeted white to the walls. I felt like a queen in a fairy story and could not ask for more...'
this bedroom from Belvoir featured on the Timothy Everest site here seems to be the paper Diana and the Duchess worked from. What one can do with a little ingenuity and a family Castle. (here)