When a player gets Rummikub & the round is over, how is play continued?
The player who got Rummikub is the winner of that round, and that round is over. In the beginning of the game, you decide how many points are needed to be the winner of the whole game.

From this PDF of Rules:


  Object:
  Be the first player to eliminate all the tiles from your rack by forming them into sets of runs and groups. Try to keep as few points in your rack as possible. At the start of the game, you decide how many points to reach to end the game.
  
  Scoring:
  After a player has cleared the rack and called "Rummikub", the losing players add up the value of
  the tiles they are holding on their racks. This score is totaled as a minus (negative) amount. The
  winner of the round receives a positive (plus) score equal to the total of all the losers' points. ...  
  
  Winning:
  The player with the highest score at the end of the game wins.


So it doesn't matter how many players there are. When someone runs out of tiles and calls out "Rummikub", it is the end of the round and you tally scores. Then everyone starts a new round (all tiles reshuffled and redrawn), and you play as many rounds as it takes for someone to reach the target amount of points.