Hi Maggy
Welcome to the forum, sorry about the delay in answering but finding a photo of the Nelson Hotel is proving to be difficult, as the hotel was demolished in the early 60's and replaced by the Fish Market (see image attached photo 1). As you will realise photograph in the 1860's was in it's infancy and good photos of that period are a rarity, but if you say you have seen one then there must be one around somewhere, so I will continue looking.
There is just one thing I have at the back of my mind I seem to remember seeing a photo where the hotel that became Oswald Baileys on the corner of Moot Street on the Bull Ring had been renamed the Nelson Hotel though I can't seem to find that photo either, but if my memory is right could it have been that one you saw?
Or could it have been this drawing of a coach outside Nelson House that you were referring to (photo 1), as from 1860 it was more likely to be a drawing than a photo.