Technically-speaking you could do it. There is no language preventing the multiclass in either UA or Dragon 312.
That's not to say it's a good idea though. The fluff very heavily implies a no on this since all three classes are meant to do pretty much the same thing: Evil variant paladin.
Why would you be curious about the alternate paladins becoming a prestige paladin? The PrC very clearly states the character has to be Lawful Good, which none of the variants are. Thus, even if, the Paladin of Freedom was allowed in the same game as a Prestige Paladin, it would be impossible for a character to be both.
If you mean whether they could be used in the same setting at all, it depends on how you interpret the foreword about prestige paladin et al. The big sentence to look at is:
If you use any of the variant classes presented here, the standard version of the class should be unavailable.
The wording doesn't take into account the possibility of using the PrC alongside the variant paladins, but that's to be expected since UA is a conglomeration of variant rules that are meant for the DM to pick and choose at his discretion. However, the intent is very clear: If you're using the prestige paladin, the regular version is not available. Thus, if you were going to use a variant on the base, you really ought to write up a prestige version instead to keep things even.
