there have been several "London Bridge" through the course of the past 500 years, some had houses, some had water wheels

the London bridge today, is in Lake Havasu, Arizona (I've posted about that before) was replaced with the Tower Bridge

in 1753, there were both houses and waterwheels. At least 4 of the arches had waterwheels that pumped water into the city water system, the below are engravings from 1749

https://thames.me.uk/s00049c.htm

https://wellcomecollection.org/works/b8ud3g79/items

See Edward William Cooke's art for a variety of images of the early and late 1820s bridges

When the bridge was replaced in 1822, its new design precluded the use of waterwheels and the responsibility for London’s water supplies was transferred to the New River Company