If I were in charge of red Alert 4, I'd charge full ahead with the crazy. The explanation is that all of the time travel has been messing with reality, resulting in psychics, super-science and all the other shenanigans.
The Allied campaign starts with a FMV of someone watching the news showing Labor has won control of the UK under an OC communist leader. Suddenly there is shaking, and you look out the window to see the UK invading the east coast. In the second mission California declares for a communist revolution and the US is now stuck in a two front war. It is quickly revealed that the Soviets fleeing the allied victory in the Red Alert 3 timeline passed through a fracture in time and landed on our moon, and began to secretly support various communists in order to prepare to take over the world. Eventually OTL allies catch up with the crazy and not-Elon-Musk builds an Orion Nuclear battleship and you invade the moon.
The Japanese Campeign follows the Japanese victory timeline in red alert 3, when suddenly they are attacked by cybernetic dinosaurs. It is Yuri from the soviet victory timeline in Red Alert 2, returned from the distant past. There are sorts of psychic shenanigans foreshadowing oncoming time-quakes, making victory bittersweet.
The Soviet campaign starts after the Soviet victory for Red Alert 1, when suddenly there is a rebellion. It is quickly revealed that the revolution is being supported by survivors from the Red Alert 2 allied victory timeline that got destroyed with Einstein's assassination. Einstein prepared a special chrono-fortress to defend against time-travel after he discovered evidence of the timeline merger in Red Alert 2. During this campaign there is a Giant Ant's mission, and the ending includes a hook for an expansion pack.
In the expansion pack, time-quakes create portals which cause each of the timelines from the base game to start fighting each-other, until they realize they need to band together in order to survive the worsening time0quakes and stabilize their respective timelines. This includes more crazy, including a very confused Scrin incursion wondering why there is no tiberium, an attack by the spynx while in egypt, and cuthulhu rising from the depths of the ocean.