I'm partial to the meteorite/natural disaster idea. And, if I could get my way, this game would take place after Godzilla vs. Destoroyah
Here's another less thought out idea,
Think of this as the opening cut scene.
Showa Godzilla is battling Showa Ghidorah. At the bottom left side of the screen it says,
"11:59PM
December 31, 1998"
Fade out to Heisei Godzilla battling Super Mechagodzilla. At the bottom of the screen it says,
"11:59PM
December 31, 1998"
Fade out to FW Godzilla battling a Kezier Ghidorah. At the bottom of the screen it says,
"11:59PM
December 31, 1998"
Now cut to the dropping of the ball in New York city, zoom in on the countdown to New Years 10 seconds to go.
Cut to Showa Godzilla charging an Atomic Ray, then to the same for Heisei and FW. At the last second of the count down, the Godzilla's fire their rays as the opposing monsters fire theirs and create a simultaneous beam clash. The beam clashes occur at the same time and create a wormhole of sorts and the different continuities mix into one where all the kaiju roam around the Earth duking it out with each other.