Well some of the streets next door in that census district are still there; enumerator went from Calkeld Lane (now an alley sloping down from Church St to Damside), then I suspect, turned left to do the wonderfully named Crooked Billet, then your Under Gardens. From there he went to Union Square and Fleet St, which I think make up today's Fleet Square, i.e. the bus station.
So I think that Under Gardens was in or around the bit of today's Damside Street that is opposite the bus station. Behind that strectch of Damside is a steep incline up to Church St, so the people living there would be 'under the gardens' at the back of the well-to-do Church St properties.
Thats my theory anyway! I've got several Lancaster history books, but none with a detailed plan of that area before 1850. If I go into town this weekend I'll take the camera, as we usually park where I think Eli used to live

sorry Eli

Barbara