If the Manhattan Project had had better security, there will be a number of effects. That is presupposing that the better security arises from both the British and Americans preventing their Soviet agents taking part. Initially this will slow but not stop the Soviet acquisition of the bomb, once the first one has been used, and it’s obvious it works, its then just a case of doing the hard work of building one. And surprisingly it’s not the building of the early bombs that was the problem; it was the making of the components that was difficult.
Now depending on a number of factors, post war it will be a race between the British and the Soviets as to who is second to build a working bomb. Both nations will be hampered by the loss of the information that Klaus Fuchs provided. When he was arrested in Britain, he handed over all the notes he had made for his soviet masters to the British, which was very helpful to their bomb project. However this in the case of Britain could go one of two ways. Without the revelations of how British spies had passed intelligence to the Soviets, Anglo-American cooperation could have lasted longer, thus resulting in an earlier British bomb. Or the Americans could have as they did, cut off all information and technology transfers, so slowing the British project.
Ether way developments in atomic weapons will be slowed, as without the fear of Soviet weapons, there will not be the push to develop more advanced weapons in the US. Also the nature and conduct of the “Cold War” will in and off itself be changed. Might the Americans, if a Korean War breaks out, authorize the use of A-bombs against Chinese targets, safe and secure in the knowledge that there can be no Soviet retaliation. Much will depend on when; the Soviets finally acquire their own weapon.
In addition, for there to be better security of the Manhattan Project, this presupposes that both the British and Americans have better penetration of Soviet espionage efforts pre and during the war. This too will have a major effect on events post war, leading to greater Soviet paranoia, and greater western confidence. Even to greater penetration of Soviet secrets, as without the information from their agents in the west, they will have a much harder time discovering western agents in the east. Will there be a period of “McCarthyism” if the majority of Soviet agents have already been exposed/turned/arrested?
What at first appears to be a simple question, with a simple answer, is in fact highly complex, and has the potential to see major changes in the post war world.