The 0.6 to 0.7 µV steps after 4 seconds corresponds to the simple calculation:
With 160K and 100 K at the ADC the theoretical range would be up to 16 V. This is divided by 6.25 MHz and 4 seconds which results in some 0.64 µV steps.
In theory there may be a little extra resolution from looking at more start / stop values (a bit like higher order filter), but it seem to be not using this and the rather slow (200 Hz) modulation does not make this very attractive.
I would normally call this 7 bit resolution, not much better.
Similar I don't think 125 ms would really give 6 digits. I should more like takes 250 ms for digits.
Theoretical ( use the root of 12 factor) 0.64 µV quantization would correspond to 180 nV RMS noise - if this would be simple quantization with no other noise. However here it is quantization at the start and stop at least, so 250 nV expected quantization noise alone after 4 seconds. So some 10 seconds integration to get 100 nV quantization noise (still more than 100 nV steps !).
The CS2000 may be used for jitter removal for the 50 Hz mains clock, but the jitter specs I found (~60 ps) are not that great compared to a low cost crystal oscillator (e.g. 1 ps range). Jitter specs are still a bit tricky, as there are different conditions / time windows used. There would be additional jitter from the logic anyway, but better than 10 ps jitter would be nice for the clock. It may be a good PLL, but even with a very good input signal it is hard to get good jitter specs for the much higher clock. The importance of jitter scales with the square root of the modulation frequency. So jitter is much less (e.g. a factor of 30) important than with the fast modulation MS-ADCs. So the PLL may be just good enough to be not a mayor problem. Chances are the rather slow modulation is chosen because of jitter from the PLL.
The effective input noise from jitter should be square root of the switching frequency (4x200 Hz) times the jitter amplitude, times 16 V (theoretical range). So this would be some 27 nV/sqrt(Hz) and thus still OK. a poor mains signal may still add jitter, though the PLL (seems to be digital) could be quite good in suppression here and may not add much extra noise.
An easy to handle noise source are the resistors at the integrator. As the noise adds as current noise the 100 K would act like about 256 Kohms in series to the input. So a total effective resistance of 416 Kohms or some 83 nV/sqrt(Hz) from the resistor Johnson noise alone.
The OPs (LTC1150) low frequency noise is multiplied by a factor of 2.6 4.16 due to the resistor ratio. So this would be some 350 nV/sqrt(Hz) 220 nV/sqrt(Hz)- a major noise source. So it could be worth using a lower noise OP here. They are available, with more choice with a low supply voltage, which would be sufficient.
The overall input referred noise sources should be at some 400 300 nV/sqrt(Hz). For 4 s integration the BW is 1/8 Hz. So the white noise would correspond to some 140 100 nV_rms. So at 4 seconds the quantization noise may still be dominant to the other noise sources. Less noise from the OP would mainly be effective a longer integration, like 16 seconds or more.
The higher frequency noise of the OP27 and the LM311 comparators would compete with the quantization noise as they have the same dependence on the integration time.
It is a bit hard to estimate an effective BW and noise for the LM311. The bias current suggest a noise level in the 10 nV/sqrt(Hz) range. For the BW I would estimate some 10 MHz, maybe a little more.
This would give a high frequency RMS noise of some 30 µV. The slope is 10 V / 100K / 100 nF = 1 V/ms.
So the 30 µV would correspond to 30 ns_rms for the comparator timing. With 6.25 MHz the quantization noise is at some 46 ns_rms and thus comparable. Reducing the comparator noise would thus be only partially effective, as there is still the limited quantization.
So for 4 second integration I would expect the main noise sources to be
1) quantization: ~ 180 nV_RMS
2) LM311 ~ 120 nV_RMS
3) LTC1151 ~ 125 80 nV_RMS
The LTC1151 noise gets more important for longer integration, less for shorter integration.
For a board remake, the question is, if this is meant only to be used in the transmille DMM or possibly also "standalone" with maybe a simple extra front end (e.g. buffer only).