The main British contribution was in spurring on American interest in the program early on. The US project the begun in 1939, after Einstein's letter to Roosevelt, was very small and not very successful. It did not generate sufficient enthusiasm among the major scientist-administrators who were responsible for choosing which scientific projects, out of many candidates, should be funded and pursued for defense purposes. The British program, however, had made calculations that made a nuclear bomb seem much more attainable for the United States — and Germany — than any previous ones had done. They sent these calculations as a report (the MAUD Report) to the US in the summer of 1941. When it got no response (the person who received it just filed it away), they sent an emissary to talk directly to major scientist-administrators in the US about its conclusions. This radically changed their view on the matter, and they began to accelerate the project. Within a year they had concluded that a bomb project was feasible and authorized the US Army to begin production work.

If the British had not gotten involved in the summer of 1941, would the US had pursued it in time to be successful? I doubt it. The prevailing view was that the risks were high and the chances of success were low. This, incidentally, was what the Germans also concluded when they, at about the same juncture, decided not to pursue a bomb project (they instead settled on a pilot-scale reactor research program, orders of magnitude smaller than the Manhattan Project). In retrospect the British estimates were over confident, and it was much harder than they expected. So in this sense, somewhat ironically, the British were crucial because of the incorrect data they gave the Americans. But once the Americans realized how difficult it truly was, they had already invested very heavily in it and their leadership wanted to push it to conclusion — at whatever cost — anyway. It is very easy to imagine that without the MAUD Report, the US program would have just languished, certainly for enough time to insure that no weapons were to be used in the Second World War.

Now, after that contribution, what else did the British do? Once the Manhattan Project was rolling (1942 onward) the leaders of it (all Americans) did not feel they needed the British to be involved any more, and actively sought to exclude them. They feared that the British would not meaningfully contribute to the making of the atomic bomb for the war, and thus were only interested in "piggybacking" on American work that would be significant militarily and industrially in the postwar. General Groves and Vannevar Bush both lobbied FDR to exclude the British, at the same time that Churchill was lobbying FDR to include them. Churchill prevailed and the Quebec Agreement of 1943 authorized full collaboration with the British.

In practice, it was less than full. Groves allowed the British to contribute in primarily two areas: bomb design work at Los Alamos, and gaseous enrichment at Oak Ridge. The British also maintained a reactor research laboratory in Quebec, but it was mostly a one-way information flow (the Canadian research could be given to the Americans, but American research was largely not given to the people in Canada). All of this was a very deliberate attempt to compartmentalize them out of several key areas of the project, including Hanford.

The British sent over delegations of several dozen scientists to help in the work in the United States, separate from their workers (British, Canadian, and — to Groves' constant irritation — French) in Quebec. Some of these members are justly famous as scientists: James Chadwick (discoverer of the neutron, head of the delegation); James Tuck (major contributor to the explosive lens work); Klaus Fuchs (major mathematical physicist whose work was key to the implosion design and who held the patent on gaseous diffusion); Otto Frisch (co-discoverer of nuclear fission); Niels Bohr (whose major contribution at Los Alamos was discussions about nuclear weapons control in the postwar, but who also helped work on neutron initiator); William Penney (whose experience with the bombing of Germany helped with developing the tactical plans for the bombing missions, and who would later be part of the damage survey teams in Hiroshima and Nagasaki); to just name a few. They contributed in a number of key ways in the development of the bomb that the Americans ended up building, in the end, and is outsized given how few they were relative to the others.

To say that they contributed a disproportionate amount, however, is not to say that the Americans wouldn't have been successful without them. It's impossible to say, of course. But the US had a lot of important research talent, and there are very few scientific members of the project who were "indispensable" in the sense that if they weren't there, you cannot imagine its success. (Among the only ones who most historians and project participants consider in this category is Enrico Fermi. But even Oppenheimer was probably dispensable, despite his impact.) A project such as this does not rest on one, two, or seven a dozen people. It was a mammoth endeavor, and if people had been absent, there would have been others who could have potentially filled the gaps.

But it's again not a thing one can assert with much certainty. We cannot replay the Manhattan Project as a simulation, minus the British delegation. But I suspect that Bush and Groves were correct in assuming they did not need them. But they certainly were able to use them to good effect — I consider the British contribution significant even if I am not saying that it was absolutely required for the bomb to have been developed.

There are, of course, other consequences to the British involvement. Penney went on to head the UK's own atomic program after the war ended and the US once again stopped working with the British (after the Atomic Energy Act went into effect). The first British bomb was very deliberately modeled on their wartime work. And Fuchs, of course, was later revealed to be a Soviet spy — to the great dismay of the Americans in the Cold War.

On the British contributions, Szasz's British Scientists and the Manhattan Project is the most concise volume. Lorna Arnold and Margaret Gowing's official histories of the British nuclear work are the more in-depth approaches. All of these approaches are rather Anglo-centric, as one might imagine; one needs to balance them against the American-centric approaches to find a medium ground.