Hello
Have you tried a page by page search of the Workhouse Infirmary in the 1891 Census, rather than a surname only search?
A few 1891 Census, only have England as a Birthplace.
The Workhouse Infirmary described as (one of the largest in the World).
The Workhouse Infirmary and grounds were reported to have covered 43 & 1/2 Acres and designed to hold 3,000. There appears to be a separate house and cottages for the staff
At the opening in 1889 the following were named in the newspaper with their former places given:-
Mr Hardie from Scotland, Administrator at Greenock Poor House and Asylum
Miss A. F. Gibson or Miss A. E. Gibson, Matron, Head of Infirmary at Brownlow Hill, Liverpool.
1891 Eccl. Parish All Saints, Birmingham
Thos Martin Hardie, Married, 42, Master of Infirmary, born County Durham (about 1849)
RG 12 2401/135 page 1 (not checked)
Hardie's children were born Scotland.
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Does the full collection of Register Books and Record Books survive, with the residence of next of kin given in one of them, when part of the family were admitted?
Birmingham City Archives (advance Notice required to order documents) were within Birmingham Central Library.
Warwick County Records Office, Cape Road, Warwick, (because Birmingham was once in Warwickshire).
If no joy there, look under NHS collections who may have deposited records in Archives. In Yorkshire the pre 1948 records at one Workhouse got split up, with part retained by Local Authorities and the remainder were left to the NHS.
The NHS being successors of that Workhouse kept them and deposited them later under NHS records and those Workhouse records survive today as separate collections in Archives.
Happy hunting, Mark