In the Spring of 1944 Mountbatten and Auchinleck requested that 10% of their military shipments be used to ship food to relieve the famine. This provided 25 ships and 200,000 tons of grain. This was far short of the amount needed but it did alleviate the famine by reducing grain prices enough to distribute food to the poor.

Lord Wavell requested 1 million tons of food to relieve the famine but was only able to acquire about 500,000 tons by the end of 1944. At Lord Wavell's request Churchill did send a letter to Roosevelt requesting food aid for India. The relatively weakly worded letter resulted in no aid for India. Roosevelt passed the letter on to the Joint Chiefs who advised him that there were no ships or excess grain available.

I wouldn't exactly credit Mountbatten with saving the lives though The worst of the famine was in 1943 Mountbatten arrived in the middle of it and didn't really act until Wavell convinced him. Wavell did all of the footwork managing food distribution and begging pleading and threatening till he got some food. Churchill or Mountbatten could have ended the famine with a strong request to America for aid. A handy "oops we gave away all of our food please send 500,000 tons of grain to India or face a complete collapse of British forces in the Far East" would have likely generated a positive response from America.