In 1752 he was appointed physician-extraordinary to George II, and the same year was created doctor of medicine at Cambridge by royal mandate.
Two years later he was promoted to be physician-in-ordinary to the king, and he was the usual medical attendant on George II in his journeys to Hanover.
He contributed to the study of chemistry in England by his translations of the works of Georg Ernst Stahl and Herman Boerhaave, as well as his own writings and lectures.
[1] Shaw's translations or adaptations included: His original publications were: He married Frances, daughter of John Hyde of Quorndon in Leicestershire.
His daughter Elizabeth became the wife of Dr. Richard Warren, who commended his father-in-law's services to literature and science in his Harveian Oration for 1768.