Moreso in Pakistan than in India,

Former members of the British India Civil Service (ICS) were sought after and offered handsome salaries for staying on through a pre-determined transition period, or indefinitely into the future.

Edward Wakefield, an ICS officer in Punjab during withdrawl, was courted by both India and Pakistan, but turned down both. His reasoning was typical of ICS officers during the draw down, "I had spent most of my life in the service of the Crown and did not feel disposed to serve another master."

British ICS officers were offered either a handsome pension that would have placed their standard of living on par with a physician or lawyer back in the United Kingdom, or a career in the British Civil Service elsewhere with a suitable salary. Many took these generous offers.

By December 1946, there were 608 ICS men in still British India. By Indian independence day (August 15th 1947), that number was reduced to 429. There were 62 left by the end of 1947, and only 15 of the original 608 made it as far as 1952. The last British bureaucrat to leave was J.W. Orr. He was Inspector of Customs and Excise in Delhi, and departed in 1955, at the age of 45, to run a gold mining firm elsewhere.

Far from being hostile to their former colonial administrators, the governments of India and Pakistan were interested in a smooth transition, and recognized the value in retaining some of the more effective ICS personnel, especially in the revenue collection departments.

As for non ICS British (businessmen, professionals, academics, engineers) I cannot say for sure.

My information comes from...

"The Chaos of Empire: The British Raj and the Conquest of India" by Jon Wilson. Chapter 15 The Great Delusion particularly.