Answer

Asklepios, the ancient Greek god of healing. 

The bust is on the pediment over the old Bath Ear and Eye Infirmary in London Road which now accommodates an undertaker’s business. The infirmary, founded in 1837, treated outpatients with mainly ear conditions. It subsequently moved to a house in Marlborough Buildings before transferring to the Royal United Hospital as the ENT Department.

On the base of the bust there is a carved representation of a rod with a single snake entwined around it. In the ancient world, snakes were associated with healing.