what if no nation had been successful in their attempts to make an atomic bomb? Assuming the allies still win WWII...
Safe assumption, that. When the Bomb was deployed, Germany was defeated. The entire Japanese navy had been sunk, the Home Islands were under blockade, and most of Japan's cities had been burned down. Japanese armies had been annihilated in Burma and the Philippines, and the USSR was about to declare war and destroy the army in Manchuria.
...how does this effect the post war balance of power?
The first thing is, how does it affect the end of the war? Does the U.S. go ahead with the invasion of Japan, and what actually happens?
OLYMPIC and CORONET were scheduled for November 1945 and March 1946. That adds at least eight months to the war. Most probably the Soviet Army occupies all of Korea and much of north China, probably as far south as Shantung. Chiang may protest, but it will seem rather absurd. Probably the Soviets will follow the pattern in Poland, and establish a Communist-controlled Chinese government in the liberated area.
Meanwhile, British forces would liberate Malaya and Singapore.
However, the Japanese actions during this period would be horrific. The Japanese had plans to murder all PoWs and Allied civilian internees - plans which were aborted just in time by the surrender.
The fighting in the Home Islands would be incredibly bloody and destructive - like Okinawa, but on a greater scale. The collapse of food production and public health in Japan would have consequences of comparable horror. Probably between 10 and 20 million Japanese might die.
That's assuming that the Imperial regime holds together. At some point, the average Japanese is going to give up, and even the Army fanatics will run out of juice. That tipping point could come fairly soon. The completeness of the OTL surrender indicates to me that a lot of Japanese were ready to give in once authority gave the word. Without that word - it would take a fair amount of beating, but I don't believe the entire nation would fight to the death.
Why does this matter? Because it affects the America posture after the war. An additional 200,000 war dead (half again the OTL number) and another year of vicious fighting may leave the U.S. much more "war-weary" than OTL.
Without the threat of mutually assured destruction coming from the nukes, could a conventional third world war between the US lead west and the Soviet lead east actually occur?
The absence of the Bomb changes the postwar situation dramatically in one key respect: the U.S. does not have an apparently overwhelming weapon. Post-war U.S. policy was very strongly affected by that perception. It led to complacency, and to an exaggerated role for the Air Force. There was a lot of talk of "push-button" warfare, with the newly separate Air Force doing all the significant fighting. There were proposals to cut back the Navy to a convoy escort force and disband the Marine Corps. The near-total demobilization of the Army left the U.S. nearly helpless at the time of the Berlin Crisis in 1948.
If there is no Bomb, that talk never gets started, and the U.S. retains much larger "conventional" forces. That affectas the U.S. economy in various ways - larger bill for the war and larger ongoing spending.
On the other side, Stalin asserted that "atomic bombs were made to frighten people with weak nerves", but he never dared use one or risk nuclear attack. The USSR could be more aggressive in Berlin, Greece, or Iran.
Down the road: it depends on how the U.S. reacts to Soviet moves, and what effect the longer and costlier war has on U.S. politics. Will the U.S. be willing to take an international role, or revert to isolationism?
OTL the USSR tried to press its power in a lot of areas on its borders. One historian compared to a burglar trying a lot of doors to see if any are unlocked. The U.S. responded by checking Soviet and Communist pressure in many areas.
ATL - the U.S. may be too war-weary to care what happens in Greece or Iran. Or the U.S. may be less complacent and maintain more force to intervene with, and the USSR may push harder.
The USSR and the U.S. were intrinsically at odds, so I don't think the peace will last more than 10 years - perhaps not even 5. The Korean War broke out in 1950, and there was already the Chinese civil war and the Greek Communist guerrilla war. Without the fear of nuclear war to deter "escalation", the conflict is likely to reach a full war.
Another deep change is the different view of Big Science. OTL, the U.S. gave Big Science a lot of money and Big Science produced the Bomb, which was an absolutely decisive factor. Ever since then, a lot of policy has been set by the idea that Big Science was on the verge of other comparable discoveries. The initial stages of the Space Race were under that idea.