The reason the Americans were cool on British involvement in the Manhattan Project was because the British had earlier frozen the Americans out of its own nuclear weapons project, Tube Alloys. Until 1941 or so, Britain was actually ahead in the nukes race, but it ground slowly because of its refusal to work with others and its lack of resources.
If the US doesn’t enter the war directly or doesn’t do it’s own nuke program, it would probably have eventually been brought in to get Tube Alloys done. This counterfactual is quite grim, because it likely ends a couple of years late with the Soviets overrunning Germany and Britain irradiating the entire continent west of Warsaw to keep them at arms length.
