Henry V111 dissolved the abbeys and replaced them with 6 endowed hospitals to serve the needs of (mainly) London. Among these were Christ's, St. Thomas',
Bridewell and Bethlehem Royal Hospital (Bedlam).
In later years other hospitals were built, and following the tradition, they also used the 'Royal' cachet for prestigious reasons. Some didn't because:
a) they were prestigious enough,or
b) they adopted a high-ranking title such as University
Of the former group, St. Bartholomew's (Barts) and St. Thomas' are examples. The University Hospital of Wales is an example of the latter.
Sunderland General Hospital changed its name to Sunderland Royal Hospital
when the ageing Sunderland Royal Infirmary was bulldozed. Nothing to do with royalty, but rather nice, don't you think ?
Interestingly, the original Christ's Hospital is now a famous public school,
Bridewell became a prison/workhouse and Bethlem Royal is now the Imperial War Museum.