Old Gloucester(shire) Royal Infirmary was in Southgate Street demolished in the 1982-86 ?( today the site is home to the Bank of England) between Parliament St and Albion St on (lower) Southgate St.

Gloucester's (old) Hospitals

The county infirmary, in Littleworth, was built, in 1755, by voluntary donation; is a plain but commodious and well-arranged edifice; has an income, from sub-scriptions, and the interest of funded property, of nearly-£3, 000; and extends its benefits to a vast number of patients.

The lunatic asylum, on an eminence in Wootton, was erected in the present century, at a cost of nearly £50, 000; is a semi-circular edifice, with two wings added to the original structure; and contains accommodation, with good arrangements, for 600 patients; and had, at the census of 1861, including officers and servants, 578 inmates.

An hospital for the insane, at Barnwood, had, at the same time, 37 inmates. The reformatory school in Hardwicke, 4 miles to the SW, may be regarded as connected with the city; and had, at the census, 43 inmates.

The workhouse for Gloucester district is in St. Catherine parish; and had, at the census, 176 inmates.

Location

I reckon that Littleworth went roughly from Southgate Street, down
Parliament Street and Brunswick Road including Brunswick Square down
towards Spa Road, maybe slightly south of it and South Hamlet took up
where Littleworth left off