For a Cold War similar to IOTL, it should be ideological too. I think you need the different blocs to represent competing ideologies, otherwise its just a rehash of imperialist posturing like in 1914.
Obviously, the USA represents electoral democratic capitalism vs the Soviet Union's communism. Now IOTL Britain and Western Europe were electoral democratic capitalist and broadly allies of the USA, forming a democratic capitalist Western bloc (with many unsavoury allies in the rest of the world).
So for all the suggestions of a surviving British Empire or Franco-British Union being the third bloc with imperialism as its ideology, is European imperialism really that hostile to the USA? Would they actually be Cold War rivals or just join together to gang up on the USSR with anti-communism? As long as the Anglo-French Empire is democratic (domestically for white people) and accepts the Monroe Doctrine, such an imperialist bloc will naturally align to the US due to their shared capitalism and electoral democratic systems right? In that case you are back to a two-sided Cold War.
To get a three-sided Cold War, you need a third ideology that is hostile to both Western democratic capitalism and Soviet communism. In WW2 you had that with fascism/Nazism/Japanese militarism.
KMT China or a more powerful Maoist China is a good 'third-worldism' ideology for a third bloc. As is Islamism. Not sure about any others that could be viable enough to become a superpower.