I doubt the 1N4148 diodes are a problem. They do not have to be matched for the circuit to work properly. Fifty hertz (or even 60 Hz) is a low enough frequency that slew rate in the op-amps should not be a problem. Since you get a clean sine wave at the output of U1A, there is no problem with the input voltage divider. The only thing I can see that might be a problem is the variable resistance, RV2, being set at too small a value. Try the obvious things first, such as checking all the connections to make sure they really are good connections, and replacing the dual op-amps, U2A and U2B. Make sure U2 is getting both + and - power on pins 8 and 4 and that the power connections are by-passed with at least 0.1 μF capacitors... the usual "good engineering practices".
This may seem totally obvious: make sure the power supply is actually on. I once spent a whole morning trying to troubleshoot a prototype analog op-amp circuit before discovering the power supply was off. Back in the day, we always prototyped on the workbench with a lab power supply, separate from the circuits we were working with. It was pretty easy to turn off the power supply before making circuit changes to the prototype circuit and then forget to turn it back on.
This could also be a problem related to ground faults associated with the oscilloscope measurement. I would use a small 6.3 VAC filament transformer to obtain an isolated, low-level, sine wave to supply input to the absolute-value circuit. You can connect a 10 kΩ potentiometer across the secondary winding, connect one end of that potentiometer to circuit common, and connect the wiper through a 10 kΩ (or larger) resistor to the summing junction (inverting input) of U2A. All this would replace RV2, U1, and the input divider circuitry.
Make sure the oscilloscope probe has a "ground clip" you can connect to your circuit common. If there is a change in waveform with the "ground clip" removed, you have a ground fault. Operating the oscilloscope from a line isolation transformer could help with that.
Other than the above suggestions, I don't know what else to suggest that you can do. This circuit has always operated "as advertised" for me. The only anomaly I ever experienced with this circuit was a slight distortion near zero output that occurs at the zero-crossing of the sine wave, where one diode is turning on and the other is turning off. The kinks and flat-topping were never observed with my implementation. They are, however, an indication that the absolute-value circuit is being over-driven at the input.