The only reason Romania, Hungary, and Bulgaria allied with Germany was because Germany dominated continental Europe at the time. The Romanians were culturally influenced by France, sympathetic to Poland, and had border troubles with Hungary and Bulgaria. They only allied because they felt they had no choice, thought they could get back Bessarabia from the Soviet Union, and win brownie points to get back Transylvania from Hungary after the war. Soon after Pearl Harbor, Marshal Antonescu made a statement along the lines of "Romania is for Germany against the Soviet Union, we are neutral between Germany and Britain, we are for the United States against Japan."
Horthy in Hungary was very wary of the Nazis despite having a lot of common interests. There was alot of anti-Nazi opposition in Hungary (alone with a lot of pro-Nazi groups). Hungary had very good relations with Poland. and wouldn't cooperate on any attack against it.
Bulgaria was willing to cooperate with Germany to get Macedonia and the lands it lost to Greece. But it only reluctantly declared war on Britain and the USA, and never declared war on the Soviet Union. It sought to leave the war as soon as feasible (that it could do so safely and not be invaded by the Germans).
Originally, Germany was very wary of having Italy in the war. The generals didn't think the Italian armed forces were ready. Mussolini only joined the war because he thought it was over and wanted to be able to show up to the peace conference.
Japan didn't quite know what it wanted to do as various factions argued over what was to be done. Japan had no interests in European politics and was only concerned with East Asia and their war with China. Their earlier work with Germany was overturn when Hitler unexpectedly signed the Molotov-Ribbentropp Pact which was a 180 degree turn from his earlier anti-Communism and caused a lot of dismay to the Japanese (and Italy too.)
All these nations had widely divergent interests, some of which were mutually exclusive. The Germans always treated their "allies" in a self-serving way. It would be more accurate to call them "satellites" and not allies. Hitler was always willing to make sudden changes in policy for short term advantage and never bothered to consult or inform his allies beforehand.
It is hard to see why they would have allied earlier. If Hitler keeps to a very hard anti-Soviet policy, its possible that Germany, Italy, Japan and perhaps Hungary cooperate more. But it leads to a very different timeline as it would impose constraints on Hitler's own policy. But it is hard to see Hitler playing nice to anyone and restraining his own actions for the benefits of others. Any alliance Hungary is in, the Romanians will be out. And Bulgaria won't be anti-Russia.