Based on what I heard from friends and others doing Veteran content, you have no choice in selecting the other faction you want to quest in.
Basically you're assigned a faction and you're stuck until you finish all of that faction's zones completely. This really bugs me it takes the choice out of the player's hands. I'm a big fan of alts and my plan was to have alts in the other factions to experience their story lines. Being forced into a Veteran Zone faction means that alts in that faction aren't viable unless I want to repeat content like crazy.
I know that my main character can get Veteran exp in Cyrodil and in Dungeons but based on what others have said on these forums, neither of these seem viable (I've heard VR1 doesn't do well in the VR dungeons. If I'm wrong about this, please let me know!)
So my other option is to wait until Craglorn comes out by knowing my luck, the developers will make it impossible for anyone under VR10 to do. I know that supposedly VR1s can go here but I'm not sure if I can trust Zenimax about this since they said initially we can choose which VR zone we want to go to.
Can we get a choice in what Veteran Zone we go in, please? This was a bad decision on Zenimax's part and should be rectified.