From The Bristol Hotwell, by Vincent Waite, ISSN 1362 7759, Bristol Branch of the Historical Association, 2002 reprint.

...in 1867 the new Pump Room was in turn demolished so that Hotwell Point could be removed and river navigation made safer. Thus the spring was lost after a long and eventful history. After much public agitation and complaining in the local press the spring was enclosed and piped to a small grotto hollowed out in the rock. Here a pump was set up in 1877 and an attendant provided by the Bristol Docks committee. In 1880 Dr. Griffin wrote a warning letter to the newspapers claiming that his analysis of this pump water proved that it was not from the original spring which in any case was too far away to retain its correct temperature. Yet up to 1913 the pump was still in use, and sometimes supplied as many as 350 persons a day. Then the long-threatened pollution of the water by the river became too obvious to be ignored and the pump was closed. The entrance, blocked up by a small wooden door, can still be seen in Hotwells Road near the Suspension Bridge.

The wooden door is no longer here, but this is definitely the place.