Hi again,
I got a bee in my bonnet about this and trawled the streets in 1881 and wrote down street names and house names. I think I've narrowed Watts Yard down a bit for you though I couldn't find it's exact location.
Go to
www.old-maps.co.uk and type "st edwards school" into the search bar. This will take you to a map.
Zoom in till you can see the location of the school.
If you click on the bottom tab on the right side of the map you can enlarge it, though it's still not very clear, it helps a bit.
Just above the school you'll see Woodview Cottages, and above this Red Lion Inn. These are all on Woodstock Road.
A bit further up the Woodstock Road is Beaumont House, and above this again is Henley House. On the right, on the road between Beaumont House to the north and the Red Lion to the south there are some houses/streets/lanes. The words St John the Baptist are written over the area. Anyhow, I think that your Watts Yard is in that area.
Following the enumerator, south of Beaumont House is a Crappers Row, then Watts Yard, then Sitman's Row, then Black Horse Lane, Black Horse Inn, William's Yard, The Woodstock Arms pub, Brooks Zone, then down to the Red Lion Inn.
I've had a bit of a look some modern maps, but couldn't find a decent one. Will have another look later.
Hope this helps a bit.
