Right fielder chases a foul ball toward the out of play line. He makes the catch right at the line at full sprint. Without instant replay it is near impossible to determine. It looked like the ball was over the line, but one foot may have still been on the playing field.

First base umpire signals "catch" and out. Home plate umpire overrules and calls ball out of play.

What is the appropriate ruling and who makes the call?

That depends on where and how the BU was standing at the time. And it wouldn't hurt to know what rules were being used, if it was instantaneous multiple calls, if the umpires talked before making a final ruling, etc.

ASA rules. Men's Church league. Team member's umpiring. Fielding team's umpire at 1B, approximately 10' off the line toward 2B. Batting team's umpire at home plate. Near simultaneous calls, no conference.

The position of the ball is irrelevant to whether the catch was made in play or not. If the player still was established on the field of play then it is an out and you would award bases to runners based on the player taking the ball out of play.

It would be no different if the player had jumped into the air in foul territory, reached over the fence, caught the ball, and then fell over the fence.

Then it is the PU's call. End of story.

If the defender's foot is completely in DBT, it is not a catch. The line is considered in play.

I had a similar situation in 16u slow pitch tourny this weekend.
Drive hit toward first base. Ball hit fair but spun up in the air about 3 feet and would have landed foul before first base. First base player gloved the ball with right foot clearly in fair territory and left foot in the air above foul territory. Ball was clearly under control before the left foot came down in foul territory. My call was fair ball - there were no arguments from anyone and the call was accepted, but there was dead silence and a lot of "chatter" among players, coaches, and fans. I guess no one knew what happened or saw it before. Correct call?

Two different things - in or out of play versus fair or foul. Out of play is determined by the fielder's feet. Fair or foul is determined by the position of the ball (no such call as "foul player". In your example, can't tell if right or wrong, fair or foul. It depended on whether the ball was over fair or foul ground when first touched.
The fielder's feet and contortions mean nothing. In other sports like basketball, football, and the "out of play" situations in baseball/softball; the foot or feet determine. Again, fair or foul is determined by the position of the ball.

Danewguy, it wouldn't matter if the ball was fair or foul if it was caught by the fielder we have a live ball out, any other runners can advance at their own peril

CecilOne:
Please clarify. If the ball was caught over foul territory the ball is foul. If caught over fair territory but player momentum caused the player to land in foul territory, the ball is fair.

Fair or foul is determined by touching not catching the ball, if the ball is touched by a player in fair territory it is fair, if touched in foul territory it is foul. I f a ball is caught while in the air over fair or foul territory it is merely out and continues to be live. If touched in foul territory it is foul and dead.

It already struck the ground.

If caught on the fly, an out (ignore fair or foul). If first touched and not caught, foul.
Correct for any touch, catch or not.

both
The player position, starting or ending, means nothing. Only the ball position matters.