As I understand, specific 328P programs in an UNO & in a solder less breadboard but NOT on the circuit board? There may be oscillator issue with the xtal circuit... Perhaps due to the board being contaminated or etching problems. It takes very little to "pull" the xtal off-frequency.
Try this: program a simply 'count forever' loop with LED blink and numbers to serial out. Configure BAUD at 9600. Test on solder less breadboard. Move uC to circuit board. Retest. If the board does not work, likely a timing issue so changed the 16MHz xtal first and then break out the high-power magnifier if problem continues.
Ray