Oracle texts:
Fork
Instant
RR
Put a copy of target instant or sorcery spell onto the stack, except that it copies Fork's color and you may choose new targets for the copy.
Relevant rulings: You maintain full control over the copy of the spell regardless of who cast the original. (I think this means that the caster of the Fork spell is considered the caster of the Forked spell, i.e. "you" in the spell text).
Time Walk
Sorcery
1U
Take an extra turn after this one.
Relevant rulings: If multiple "extra turn" effects resolve in the same turn, take them in the reverse of the order that the effects resolved.
So, first player 1 casts Time Walk. Then player 2 responds with a Fork, and to make things easy let's say nothing else happens. Then player 2's Fork resolves, and finally player 1's Fork resolves. Since extra turns are taken in the reverse order, player 1 will get his extra turn first and then player 2 will get his.