[TokuKnight89]

Are we to assume that W exists in the "Prime Rider Timeline" from Ichigo-Kiva, but there is NO alternate W of Another Dimension/World?

In Movie War 2010, Dummy Dopant merely flees to the sight of the "Decade & Friends VS. Super Shocker" battlefield, which gives some credence to the idea that the majority of the Movie War 2010 fights occuring in whatever dimension which is largely considered the "Main World".

This is bolstered by the sight of an alternate Narumi Sokichi who doesn't know Shotaro but respects him, and this is the main point of evidence of there being no Alternate W. That being said, it's possible that the W of this Alternate World has someone entirely different than the Shotaro and Philip that we know as W. The hole with that presumption is that the Alternate Sokichi is in Skull form, and Sokichi Prime hates Gaia Memories except in dire situations. This could all suggest that Skull, not W, is the Main Rider of the apparent "W World". Again, however, Alternate Sokichi is summoned with a Skull Card, which could conceivably summon him in a "Forced Henshin" state (which is a whole nother can of worms in its own right).

P.S.: I say "Ichigo-Kiva" because Tsukasa's origins are vague at best. He has a childhood and a home and a family, but Wataru Prime seems to think that he has no greater story or meaning than to "save by destroying". It's not immediately clear which World Tsukasa's home existed in.