Lesley do you have any proof that he was moved to Coleshill Hall, the hospital in Marston Green, where he was a patient, was renamed Chelmsley Hospital, my son worked there in the 80s and I did volunteer work.
Most of the patients who were in there as children stayed for the rest of their lives, I never heard of any going to Coleshill.
Your uncles death is registered in the Staffordshire Moorlands , records show he died in Cheddleton.
If you say he was in hospital all his life, he was possibly in St Edwards in Cheddleton, I was born near there , it closed in 2002 and is now apartments.
Could it have been that his family moved to the area and wanted him near them.
Records are possibly at Stafford records office.
I don't think that Birmingham library would have records of Marston green/Chelmsley hospital as it was administered by Meriden and Meriden records are kept at Warwick.