were there any campaigns organised in Britain or America to have
specific individuals released from Nazi concentration camps?

I have never seen references to any 'campaign'.

However, Home Office file HO 45 23741 in the National Archives at Kew
includes a Special Branch report dated 1 November 1939 from a blanked
out informant which claims:

'It was through Mosley's wife that the Austrian Rothschild was released.'

Mosley was Sir Oswald Mosley, leader of the British Fascists, and his
second wife was Diana nee Mitford. Both Diana and her sister Unity
were famously chummy with Hitler.

The report also states that 'Mosley nipped in and pinched the reward'.

The reward had allegedly been offered by 'Rothschild in Paris'.

True or false?

Who knows? But money is often helpful for winning favours.

This campaign began in Paris, but received support in Britain and the US:

"The Multinational Campaign for Carl Von Ossietzky" (1991), including
a section on "The Campaign in England" and on "The Campaign in the
United States":
http://www.irwinabrams.com/articles/ossietzky.html

"Carl von Ossietzky ? Biography":
http://nobelprize.org/peace/laureates/1935/ossietzky-bio.html

"Biographical/historical note":
http://www.iisg.nl/archives/gias/f/10771102.html