Q: If My Turn Next is played (out of turn) in the same turn as someone discards the TARDIS, who gets the next turn, and do they both get extra turns?

…Player B starts turn, Player A puts down My Turn Next then. During their turn Player B discards their TARDIS in play for another turn. Who goes next, and do they both still get the extra turn?

A: They both get an extra turn, but, in this case, if there are only two players in the game, it’s going to look like nothing happens. If you’ve got more than just those two players in the game, the non-involved player/s will definitely notice that A and B got extra turns, no matter what order they went in!

So the order in which those extra turns play out depends on the order in which the extra-turn cards are invoked. Since Player A played My Turn Next before the TARDIS extra turn was invoked, the Surprise player gets the next turn… then the TARDIS discarder… then whoever was going to go next after that turn. If it’s only a two player game, that next person is Player A, which will feel like nothing happened. Otherwise, it’s whoever is usually after Player B, which may be Player C, D, etc…

In more detail for this situation:

(Presumably Player A’s turn was before this)
Player B starts turn
Player A plays My Turn Next
Player B discards the TARDIS, and does whatever else they may do on their turn
–> Player A gets their magical Surprise turn
–> Player B gets their TARDIS turn
The next player takes their turn.

If it’s a two-player game, Player A’s turn is next, and it will seem like nothing happened: (A), B, ->A, ->B, A… Though technically, those turns with arrows are extra, but since the two players continued to simply alternate, there’s no practical effect.
If there are more than two players, however, then (A) B, ->A, ->B, C, D, etc… and players C and D will notice that A and B got extra turns in there, out of order.

If, on the other hand, Player A played My Turn Next after Player B discarded their TARDIS, the turn order would go like this:

(Presumably Player A’s turn was before this)
Player B starts turn
Player B discards the TARDIS
Player A plays My Turn Next
–> Player B gets their TARDIS turn
–> Player A gets their magical Surprise turn
The next player takes their turn.

If it’s a two-player game, Player A’s turn is next, but the order will go like this: (A), B, ->B, ->A, A…
If more than two players, then (A) B, ->B, ->A, C, D, etc… Once again, players C and D will still notice the extra turns by A and B.