The first point to improve would be to improve the ground routing: take out the shart GND wire near the LT1001 and give the heater current a separate return path to the regulator or power connector.
The heater GND and Power follow different path. They meet almoust at power main decoupling cap (470nF).
The ground to the zenner is almoust common to the ground from the LT1001 which is buffering it, and the VZ is very close from IN+ from same buffer opamp.
I increased now the size of the ground and power lines up to the heater lines derivation to try to reduce eventual effects by adding a layer of solder of 1mm.
I also noticed that the 10nF may not be enough to the amplifier stage bypass capacitor. I think I will increase that to 100nF. I see some examples on LT1001 datasheet with it and I have a second board where the noise was close to 300uV. After much scratching my head and many tests I decided to test with outer 22nf cap in the bypass position and the noise immediatly went down, so I guess will go with 100nF.
I think I will follow iMo suggestion for the reduction of the biasing resistors. Makes litle sense to spare 1mA of biasing current when the LM399 are consuming 17-20mA to the heater.
As for the controlled temperature room to test this stuff still need to think how I can do it. The air flow of the air conditionair is not the best way to test. The best I can do is turn it off during the night and goes to 18ºC and turn on during day and goes to 24ºC.
But this also impacts the Keythly2000.