This circuit usually is extremely stable, I have designed exactly that topology, up to 100mA.
See here:
https://www.eevblog.com/forum/metrology/recycling-of-precision-current-source-noise-reduction-for-low-burden-shunts/msg1432410/#msg1432410The stability @ 6V supply seems to be reasonable, whereas at 12V, it should be the same stability, as this circuit does not depend on the supply voltage of the current path.
This excessive drift does not fit to the parameters (T.C.) of the high quality components.
Make sure, that the FET does not heat up the VPR221. Therefore separate both heatsinks.
The different batteries used, and their inter connection is also not precisely drawn in your schematic. Please provide an excact schematic, how these 3 / 4 batteries are connected to the circuit.
The reference and the OpAmp and the GND sense side should best be supplied by the same battery, to avoid any ground loops.
Please check, if you have used a correct Kelvin / 4 wire connection for the current shunt VPR 221. See datasheet of the shunt.
The current connections I+ I- should be connected the FET, and to - of the Battery VS / 6V, whereas the sense lines S+, S-should be connected to the + of the OpAmp and to GND of the Opamp supply.
There must be no other connection between GND and the - of the VS battery!
Then the OpAmp should be checked, if it's still ok.
If I remember right, a similar OpAmp, AD706 is used also in hp 34401 DMMs, and that caused a lot of failures after years, in the Ohm range, due to breakdown of the input transistors.
That was a systematic error, happening on many instruments.
So I wouldn't trust this salvaged part at all .. check if the reference voltage @ + and feedback voltage @ - are stable and always identical to a few µV, or if the difference voltage drifts.. that would imply an excessive bias current.
Maybe you choose another Opamp which has rail-to-rail voltage characteristics, or use a chopper OpAmp.
Concerning both 34470A, before you start measurements, without current, let them warm up 1h at least, and do an ACAL on both, then NULL both instruments in the 1A range.
That might improve the accuracy.
Current measurement is not so precise by specification, anyhow, so the absolute difference in reading seems to be all right.
If one of both instruments already has FW 3.0, you can use the DIG feature for DCI @ 20µs to check if the current output is oscillating.
I've also implemented a stabilizing capacitor C6 of 10nF between - and the direct output of the OpAmp, and this is very important for stability over different loads, and compliance voltages by the FET.
Frank