This happened at tonight's FNM and I wanted to get a clarification for the future since there are differing opinions with the judges at the event.

Adam and Neal are playing Modern Constructed FNM. They have finished Game 1 and are on Game 2 when time is called. Adam sees that Neal will be able to kill him on turn 1 so concedes the game to Neal. The players then shuffle and begin Game 3, taking their 5 turns. They report their match as 1-1-1.

There are two different opinions about the end of match procedure when time has been called.

1 - The players have 5 turns to complete their game. When Adam conceded to Neal, the game is over and the match should be reported as 1-1.

2 - The players have 5 turns to complete their match. Adam conceding to Neal on Turn 0 allows them to have turns 1 through 5 to at least start Game 3 even though it will end with a draw.

The Tournament Rules for End-of-Match Procedure are

2.5 End-of-Match Procedure
If the match time limit is reached before a winner is determined, the player whose turn it is finishes his or her turn and five additional turns are played in total. This usually means that one player takes three turns and the other two, but a player taking additional turns may affect this. Team tournaments featuring multiple players playing together (such as Two-Headed Giant) use three turns instead of five.
If the game is incomplete at the end of additional turns, the game is considered a draw..

As this is FNM I know that it is only Regular REL, but I am hoping to also get clarification at all RELs.

The match should have been reported 1-1-0. If a game ends after time is called, a new game can't begin unless you're in a single elimination round. This is a tournament rule, and applies at any REL.

Additional turns only apply to a game in progress, not to the match. A game begins when both players keep their hand - i.e., after mulligans are completed.

So, as Josh said, the match ends at 1-1, and no third game is played.