The diodes inside the opamp loop make a sharp full-wave rectifier. It may not be dead-accurate in the upper audio octaves or down around the bottom stop, but more than good enough for speech/music VU-ing.
It does nothing about meter dynamics. If the meter over-shoots bad, you can play with 10uFd-200uFd across the meter, but "dynamics" is why you should use a for-purpose meter.
A full VU+3.9K is about 7500 Ohms. I implemented this real simple. The ratio of this to R117 R118 multiplies the current 10X or 20X to your 1mA area. Note the polarities at the opamp inputs (I flipped one to make the topology clear).
Power is a 100K 2 Watt resistor from the raw 350V DC point, to two 12V Zeners, to ground. No more filtering needed.
Opamps should be LM324 types. Need low supply current and to-gound inputs. LM358 is the dual of quad '324. Costs less than a postage stamp. The power supply is designed for the demand of LM358 plus meter, don't substitute without deep thought.
This will work if the opamp inputs are above zero V DC and below 20V DC.
So to make it work as VU, we must bias the inputs up into this range. 10uFd 50V N.P. caps and 100K resistors to +12V do this.
You should calibrate R117 in VU mode to read correctly on steady sine wave.
For the "+10 range" the 7.5K and the 6.8K make a 6dB (5.8dB?) pad as designed.
For "GR" we need to sense a voltage-difference across a 7500 resistor bridged across the Neon+LDR bridge. It works out that both nodes are at several Volts DC, within the opamps' input range.
Once the R117 is calibrated as VU, it should not be changed when calibrating the GR bridge. Zero Adjust should happen fine. LA2's R25 is a trim for the audio-chain's attenuator resistance (not 68K, but 68K||100K||100K=28K), plus/minus the difference between the audio LDR and the meter LDR. As they may be 50% different, R25 may have to be quite far off 28K to get -20dB GR to touch "-20" on the meter. (May be best to calibrate for -10dB GR, and let -20 fall where it will.)