I remember the water tower at Highcroft Hospital, it was in the vicinity of the laundry. The tank on the top was a Braithwaite tank, they were a sectional bolt together tank to the size you want.
There were quite a few of them around Birmingham, there was one on Goosemore Lane for the laundry too.
The hospital may have had their own well and pumped from that. Most certainly a hospital would need a reliable water supply. Around the time the workhouse was built, there may not have been a constant water supply in Birmingham.
The early water supplies were not constant and often only operated on certain days or over limited hours of the day. Therefore, lots of houses and factorise had a water storage tank in the roof or up a tower. This would allow the storage volume to build up and even out demand.
The problems with this intermittent supply were when the water was cut off there was a tendency for polluted ground water to be sucked into the water supply pipes. People were still useing cesspits, therefore constant water supplies eventually became the norm.
There were of course problems associated with water tanks in lofts too. Dead birds and rats often found their way into these tanks and could cause problems. This is actually the reason why mixer taps were not very common in the UK early on; they were seen as a possible source of mains water pollution.
Most older houses would have a single drinking water tap on the kitchen sink. The other cold-water taps being supplied by the cold-water header tank in the loft. This tank then fed the hot water cylinder too. Its why your parents would say don’t drink form the hot water taps.
I did a lot of work modifying water supplies to houses and multi-story blocks of flats around Birmingham.