The only significant reason for the deployment of the Boeing Washington - for the four years in question - is only that the Lincoln was designed for use with conventional bombs; its bomb bay was therefore long and relatively shallow, compared to the squat, deep bomb bays on the B-17, B-24 and B-29 - the very reason why the RAF did not like the American aircraft for bombing ops in WW2. They could not carry large bombs. For a nuclear bomb, however, the Lincoln's bomb bay was rather the wrong shape, and it would have ended up hanging outside, much in the way Grand Slam bombs did on B.I Specials. Not ideal for long distances. The Washington, as shown over Japan, did not have this problem.
To come back to it, No.90 Sqn RAF was the first offensive user of the B-17c/Fortress B.MkI (8th July 1941, against Wilhelmshaven), before the Americans joined the fight, and while Boeing had marketed the aircraft as able to out-fly (especially in terms of ceiling) and out-fight the opposition, this turned out not to be the case. The USAAC had, however, neglected to read the manual, and considered that Fortresses should be used in mutually-supporting formations. By then, the RAF had decided that the Fortress was really not what it needed for bombing operations, and rather than scrap the things, inflicted them on Coastal Command instead after they had made a brief and again unsuccessful tour of Egypt.
The only further use made by Bomber Command of Liberators MkVI et al, and Fortresses MkIIa and III was as radio countermeasures aircraft, with No.100 Group; here, their cavernous bomb bays were modified into R/Os offices, and equipped with all the latest toys.