The RAE at Farnborough took an interest in using rockets for "non-projectile use" from late 1939 initially to assist in getting heavy bombers airborne before switching to the possible use by the RN. Practical trials began around April 1941 (a Blackburn Shark was an early guinea pig for the trials) and continued into the mid-1950s when steam catapults began to be the standard fit on carriers.A long time ago (early- to mid-1990s?) I ran across a mention and photos of tests done on a Wildcat with RATO. IIRC it was the British or Canadians doing the testing.
Heavy bomber interest waned in Aug 1941 after a 4 engined Stirling was written off after the rockets all fired together from an underwing mounting in an airframe not designed to cope with the sudden stress. All this in full view of an audience that included high ranking RAF officers and a cabinet minister! There is a photo in existence of an earlier successful rocket assisted take off on that aircraft.
RN interest continued because British aircraft types were not compatible with the catapults on the US built escort carriers beginning to be made available under Lend Lease and because British built escort carriers (except Pretoria Castle) and MAC ships were not equipped with catapults at all. So it saw operational use on Swordfish (especially the Mk.III in 1944/45) & Barracudas. While I don't recall seeing photos of US types with it, I've no doubt they were trialled with RATOG at Farnborough at some point in WW2.
Barracuda II making a rocket assisted take off from the very wet deck of HMS Trumpeter