Studio support seems really spotty. Only 10 out of my 47 UV items came over, the rest are geyed out.
But yes, assuming that the movie you purchase with Verizon has the UV logo on it (for example, 22 Jump Street), that movie will show up on all UV linked services after purchases (with the general caveats about studio support for that particular service.)
22 Jump Street is available on Vudu, so if you have your accounts all linked to the same UV account, purchasing 22 Jump Street from Verizon would mean you could then watch it on Vudu. Vice versa would work as well. If you purchased 22 Jump Street via Vudu, it will then show up in your Verizon purchases.
Now, you would have to be completely off your rocker to buy 22 Jump Street from Verizon though since it's $19.99 from Verizon and an HDX from Vudu is $17.99. However, it's possible that Verizon could have some specials on pricing that are advantageous. Also, if you happen to have a Verizon box on a TV that doesn't have smart app support, you could now watch your movies from it.
At the end of the day, it's just nice seeing more support for UV so it can fulfill the role that was intended for it.