The relevant part on the historical hospital from the Wikipedia page:
The hospital was described as ancient in 1215 and was named after Thomas Becket — which suggests it may have been founded after 1173 when Becket was canonised. However, it is possible it was only renamed in 1173 and that it was established shortly after St Mary Overie Priory was founded at Southwark in 1106 .
Originally it was run by a mixed order of Augustinian monks and nuns, dedicated to Thomas Becket. It provided shelter and treatment for the poor, sick, and homeless. In the fifteenth century, Richard Whittington endowed a laying-in ward for unmarried mothers. The monastery was dissolved in 1539 during the Reformation, but reopened in 1551 and rededicated to Thomas the Apostle. It was reopened through the efforts of the City of London who obtained the grant of the site and a charter from Edward VI and has remained open ever since. The hospital was also the site of the first printed English Bible in 1537.
At the end of the 17th century, the hospital and church were largely rebuilt by Sir Robert Clayton, president of the hospital and a former Lord Mayor of the City of London. He employed Thomas Cartwright as architect.