IIRC about 80% of the entire Manhatten Project budget was spent on buying the land, and building the base at Los Alamos, the total cost for the bomb itself wasn't all that great in comparison.
This is why nations like Iran and North Korea can build atomic weapons, because the key problem is getting yourself set up with the staff to produce the enriched material, and developing the detonation method in the first place.
Arguably Britian was one of three nations that could have produced an atomic bomb within the war years, assuming that generally everything is much the same as it was. This is because Britian had the expertise, and had access to the materials and resources needed. What Britian didn't have, was serious political backing. This is why Tube Alloys had as small a budget as it did, and why later on the British effort was added to the American effort...because that's where the money was...¬.¬
Therefore, what you really need is a British version of the famour letter.
POD:
So instead of Szilard writing to the American president, have him write to Churchhill or King George with the warning, reminding either that London is within bombing range of Germany, and if the Germans have this weapon before Britian, then Britian cannot hope to prevail.
This sets seeds in motion.
Key point: Come 1940 with the Battle of Britian now being fought and the Blitz falling on London, you now have a very real threat, that was stated in that letter. Therefore you are going to have a frenzy of activity to 'get it done'.
Key point: Stalin wanted a second front in '42. Well the Brits can't exactly give Stalin that, but what they do know is that if they can get the weapon finished, they can use that instead, plus it puts Stalin in his place.
Really, that's the kind of thing it requires. What many people forget, or don't want to believe, is that the Manhatten Project doesn't represent a pinical of achievement. It simply represents when there is political backing to science, wonders get acomplished.
Another example is going to the moon.
Sure there are costs involved, but the biggest 'cost' is expertise. If you don't have that to begin with, that's when costs cost. If you do have it, its more a question of getting the people together then asking them what do they need and letting them get on with it.