When did the British Navy stop awarding prize money to crews? Did the US Navy ever engage in this practice?

The RN stopped the practice surprisingly late - they were still handing out prize money well into the 20th Century. Prize money was given out to crews who captured or sank ships in the First World War. An example announcement for such a distribution comes from the London Gazette of 31st March 1916:

Notice is hereby given to all persons, interested therein, that preparations are now being made for the intended Distribution of the Prize Bounty awarded for the destruction of the enemy armed ship "Cap Trafalgar" by H.M. Ship "Carmania," on the 14th September, 1914.

Agents or other persons having any just and legal demand, unliquidated, against the Award are required to transmit the particulars of any such demand to the Registrar of the Admiralty Division of H.M. High Court of Justice, in order that the same may be examined, taxed, and allowed by that Officer, and paid under the sanction of the Judge of the said court. Due notice will be given, by future advertisement in the London Gazette, of the date proposed for the commencement of Distribution.

Such awards were also made to submarine crews who claimed sinkings. Crews could also gain money through salvaging flotsam and jetsam, and this was a common bonus for those of non-combatant vessels like trawlers and tugs.

Prize money was also collected in the Second World War. However, as the RAF had made a major contribution to naval warfare, part of the money went to them. Out of a total sum of £5,250,000 (in 1945 pounds) available in prize money, £1,250,000 went to the RAF. This money was not shared out to RAF servicemen, but instead was paid to "welfare and charitable organisations serving present or past members of the Air Force, including officers and airmen from the Colonies". The portion of the money paid to the RN was not paid to any particular crews, but instead shared out between all those in the services who had served at sea for six months. This included Royal Artillery gunners serving aboard merchantmen, and merchant seamen who served aboard ships chartered by the RN.

Prize money was also paid out by the USN in the 20th Century, with the crews of USS Omaha and USS Somers receiving money for their capture of the German blockade runner Odenwald (which had been disguised as an American ship) on the 6th November 1941. This was the last payout of prize money made by the US Navy.

Sources:

http://www.navalhistory.org/2011/11/06/neutrality-patrol-seizes-german-prize-6-november-1941

http://www.naval-history.net/WW1NavyBritishLGNavalPrizeMoney.htm#lg

Hansard - Prize Act 1948, Naval Prize Act 1928