The black and white shield has been the badge of St Bartholomew's Hospital for more than 500 years. Originally the Coat of Arms used by John Wakering (Master of the Hospital 1423-1462).
Although this Coat of Arms was not granted to John Wakering or his family by the College of Arms, it became associated with the Hospital and continued to be used after his death. 

The shield appears as the Arms of the Hospital in a 16th Century roll in the College of Arms and, though never formally granted to the Hospital, continued use since the early 15th century presumably established the right to display them. Can be seen in the book of public arms, 1915.
The Book of Public Arms, Arthur Charles Fox-Davies, 1915
John Wakering