I highlighted the key idea here.
If the United Kingdom does spend the resources to develop the atomic bomb on its own and a delivery system, then they would not have the resources to build other things. The result of such reallocation of resources could result in a far worse fate for the United Kingdom. While it is popular for folks to ignore these unintended consequences, these are real concern--resources are scarce and decisions have consequences, many of which are unanticipated.
Agreed.
Baring other butterflies that help the UK, the UK will be in much, much worse shape. Lets take two scenarios:
1) Starts in 1934, after patent is approved. The most likely place to find the money is the existing R&D budget for the military, and the second most likely is diverting resources from university hard science budgets. So for this part, a good way to simulate would be to delay the development of technology across the board by 2-4 years. Imagine the Battle of Britain fought with the 1937 radar network. Or the battle of the Atlantic fought with two year older sonar/radar. Another great place to find bright young minds is the teams breaking the German codes.
Now at some point the program goes from pure R&D to applied R&D. It will now need a lot more funding than the R&D can provide. So we are now looking at delaying canceling ship, tanks, and airplane budgets combined with fewer active regiments and squadrons. This is harder to model, but one scenario might be two weapons in late 1943 in exchange for a 10% reduction is size of UK military using that introduces equipment 1-4 years behind OTL. And once France falls, which will be even easier in this ATL, the UK likely defunds the Tube Alloy.
2) Some have suggested a latter start. So lets say a start after Munich. It is hard to see where the resources would come from and how the bomb could be completed before 1947. But lets say the decision was made, and for simplicity of modeling, it consumes 10% the military budget of Canada, which should be about the right order of magnitude. So in 1938, Canada uses its vast hydroelectric resources and uranium mines and people to start working on the bomb. Even with this huge commitment, the bomb is still after the war. The Allies are short 130,000 men and about 10% of the overall budget. Initially it would not be that bad, only losing a division. But for example, Canada produced 800,000 trucks, so what it only produces half as many? Someone is losing a lot of mobility, quite possibly the Russians. Ships = 400. Bomber command personnel = 1/6 of UK forces and even higher % of training bases. Hard to model exactly, but clearly the UK is net harmed.