Ray
Yes to Vauxhall Rd where the Ashcroft Estate is now, a cavalry Barracks that was closed in the 1930's. Yes to Garrison Lane I believe there was a barracks off Garrison Lane and Little Dartmouth Street that closed in the late 1700's it could have been possibly the area where the park is now.
Alum Rock & Washwood Heath and Alum Rock I'm not sure of, but most areas had TA barracks so I don't doubt that there were barracks there at sometime.