Hi JanA, welcome to the forum.
I can confirm cows and horses were kept in the 32 acre grounds of Moseley Hall, during it's time as a children's hospital and it was one of the few things I remember about my stay there to have my tonsils removed in 1952-53. They were there possibly up until 1967 when the hospital was closed as a children's hospital and reopened as a geriatric hospital after a new ward block was built.
There is still a building that remains on the site adjacent to the Dovecote, known as the Cow House.
https://moseley-society.org.uk/index.php?option=com_content&task=view&id=17&Itemid=31
The building you would have been in, is still in use as an administration block, and is known as "The Old Hall".
How do I know, I worked there for 28 years.
I have moved this post to the Hospitals thread, where you will see a number of references to Moseley Hall
Colin