But where is the quote that says Undead Legions can be used in regular games of Warhammer?
I don't have the Storm of Magic book, so I could be mistaken, but I believe it explicitly says that the SoM rules are optional and should be agreed upon before the battle with your opponent. There's no such clause in the Nagash book. It has rules in it. Period. If it doesn't specify that they are not official, then they are.
Page 4 "How to use this book" is the closest it comes to a difinitive answer "this book is intended to be a companion volume of warhammer:nagash" so all the rules of book 2 are just a companion book for book 1 allowing you to play out the changing battles of this very specific time. The most key part is the constant reference to "This section of warhammer:nagash describes..."
And if we really want to get into semantics of who says what is allowed when and where, the How to Use This Book page ends with the words:
"Along with all of the other sections of this book, it is intended for use both with the scenarios presented herein,
and in any other games of Warhammer you would like to use it for. As you can see, this book represents much more than just a selection of scenarios and special rules to go with them. Instead you should think of it as a toolbox, from which you can pick and choose what to use
in any games of Warhammer that you play. This is highly appropriate; as a result of Nagash's return, neither the Warhammer world, or the Warhammer battles that you fight, will ever be the same again!"
(Emphasis is mine. Just pointing out that it
does state in print that this is for everything.)
Can a vampire character on a coven throne join a unit of tomb kings skeleton chariots? If so, that may open up some interesting and likely completely unintended possibilities.
The basic rules prevent characters riding monsters or chariots from joining another unit, so I don't think so. I don't have the
TK book, but I'd assume the Tomb King/Prince characters on chariots have a special rule that specifically allows them to join Skeleton Chariot units. There's no such rule for Vampires or Necromancers on Coven Thrones and Corpse Carts.