Atom bomb carrying Lancasters, it was only a matter of time. Joe and Dave have said what's required - The Lancaster was never 'proposed' for dropping the bomb in action, it was suggested by Ramsay that it was the only aircraft that could carry Thin Man internally - Ramsay's words in a report dated September 1945:
"Except for the British Lancaster, all other aircraft would require such a bomb to be carried externally unless the aircraft were very drastically rebuilt."
In the same report he stated the following, which highlights that although the Lancaster might have been considered for trials, it was a proposal only and was never going to be put into practise:
"In the fall of 1943 it became apparent that plans for full scale tests should be started. In view of the critical shortage of B-29's it was at first proposed that a British Lancaster be used for the test work even though
a B-29 would almost certainly be used as the combat ship."
The Lancaster simply did not have the performance to do it, nor did anything else in service use at that time,
other than the B-29. We forget that the B-29 was a great leap in capability over what the status quo was.
Small addition to the discussion for clarification, the Thin Man bomb was abandoned because when it was first being developed it used plutonium, as we know, that element has to be refined from uranium, and it doesn't exist on its own as an element. The refining process at the Hanford, Washington State site created plutonium of a less than pure state, which jeopardised the effectiveness of whether or not a chain reaction would be created. Meanwhile, Manhattan Project scientists had toyed with the idea of using uranium 235 as the element of choice and although less stable than plutonium, it didn't require refining to the same extent and because of its lesser stability meant that the gun needn't be as long as if plutonium was used, so Thin Man effectively died when the discovery that U235 could be used instead of Hanford's impure plutonium. This is where Little Boy enters the room and U235 enrichment was done at Oak Ridge, Tennessee from March 1944 onwards, whereas beforehand, there were only Thin Man and Fat Man.
The origin of the names Fat Man and Thin Man come from Robert Serber a former student of Oppenheimer's; Thin Man came from the Robert Dalshiell detective novel of the same name and Fat Man was the name of Sydney Greenstreet's character in the 1941 film
The Maltese Falcon. Little Boy was simply a contrast to Thin Man. Air Force personnel called them Roosevelt and Churchill for secrecy reasons