The ability to play as YOUR character on the Shadowlands beta instead of a template character is now available! You’ll be able to create a copy of your character to play on the beta, as long as you aren’t a Demon Hunter or Death Knight. There are currently bugs with copying Demon Hunter and Death Knight characters.
If I had to venture a guess as to why those two particular classes have bugs associated with copies, I’d say its likely linked to the fact that those two classes start at a higher level when you create one. I anticipate that Blizzard will resolve this soon however.
Character copy is now available, however, there are issues copying Demon Hunters and Death Knights.