I have a friend "J" who will say uno when you say uno to make you draw the the penalty. Does he have to give you a chance to say uno?
From the original Uno rules:
The game continues until a player has one card left. The moment a player has just one card they must yell “UNO!”. If they are caught not saying “Uno” by another player before any card has been played, the player must draw two new cards.
It doesn't explicitly specify that you must give someone a chance to say "Uno", though it does seem to advocate doing it as soon as you have one card left.
In my experience it has always fallen under the category of house rules as to how much time constitutes a "chance". Agree on something. If you can't, maybe ask "J" how much he likes playing Uno with you and if he would like to be invited back.
In a previous response someone cited a better portion of the rules
A player who forgets to say UNO before his card touches the discard pile, but "catches" himself before any other player catches him, is safe and is not subject to the penalty. You may not catch a player for failure to say UNO until his second to last card touches the DISCARD pile. Also, you may not catch a player for failure to say UNO after the next player begins his turn. "Beginning a turn is defines as either drawing a card from the DRAW pile or drawing a card from your hand to play.
In honor of that I'm deleting most of my post in agreement with this official decision.
This first to say Uno concept is part of what makes Uno to many people a broken game. You can't definitively end the game and you can theoretically keep it going forever. I'm surprised there wasn't a Key and Peele skit on it. The rules say you have to say Uno if you only have one card and that you can call Uno to call someone out on not having a card to force a draw penalty on them. What isn't explicit is timing hence your question.
If they are saying Uno at the same time
when you say uno
then I would say the tie goes to the active player. Because I believe the spirit of the rule is about getting someone so involved in the game they forget to say it.
You could also try a varient where only the next player (or maybe only previous... or maybe only neighbors) can say Uno. that could be an interesting twist as well and also limit loud uno yellers.