I have a 1952 C.C. "Bowl-a-Ball" rebound ball bowler, but I never knew United
produced a ball bowler as early as 1951. I wouldn't mind knowing the title of
your machine..
Anyways, like mentioned, you'll want to check the "Player-Step-Up" unit, along
with the "Coin Unit". This is especially important for the "Player Reset" Relay to
energize at the proper time..
For the Player Step Up, my guess would be to check the switches in the "Player
Reset" Relay, most likely a make-break switch. The switches in the "Score Control"
Relay, and also the switches that run off the Score Motor's cam. Can't help on
which one, but rechecking them all wouldn't be a bad idea..
Before adjusting the switches, it's always a good idea to tighten' up the switch
stack screws first..
Any chance of sending a pic or two of your machine?

Yup, and I'm pretty darn sure that it's a Bowling Alley that is being dealt with here..