Atmega328P 32.768KHz Crystal Issue
The problem here is that you don't have the load capacitors on the crystal.  These connect from each pin to ground, and their capacitance is specific to the crystal you are using.

A resistor in the oscillator input may also be needed (\$R_L\$).  The ATMega's datasheet will tell you about that I would suspect.







simulate this circuit &ndash; Schematic created using CircuitLab

When you were on a breadboard the breadboard's inherent stray capacitance was providing the function, but now you are on the PCB there is considerably less stray capacitance.

The capacitors are usually in the range of 30pf or so, but read the datasheet for your crystal to get its load capacitance and use that to calculate the right capacitor values:

$$
C_{X1} = C_{X2} = 2(C_L - C_{stray})
$$

\$C_{stray}\$ is the stray capacitance of the PCB (it can probably be ignored as it will be tiny), \$C_L\$ is the load capacitance from the datasheet for the crystal.