Slightly more than 40 C is a reasonable temperature, not too hot, especially for a system without fan. The voltage drift also seems to mainly follow the measured temperature, except for the very fast initial part, which could be the LM399 reference.
Without a fan I don't think there is anything that could alter the time constant. So it's down to a rather high temperature coefficient for the +-10 V reference. I looks like they path is 7 V reference (likely negative) to the -10 V and than to the +10 V. So it might be interesting to see if the -10 V are drifting to. From the +10 V and +-10 V difference data, it looks like that.
Without a schematics for the reference board, it's a little hard to see how the -7 to -10 V conversion is done. I don't see a way the divider with two 30 K resistors in ADC hybrid could do the job (maybe if there is a mistake in the plan, having 30 K and 70 K ? - than this could make some sense). So it might be worth to check the voltage at pin 22 of the adc hybrid, going to the ref board: with two 30 K resistors this should be about -5 V, with 70K/30 K it would be -7 V.
The ADC hybrid is used to do the -10 to +10 V inversion. With so much drift (150 µV), I would not expect the external OP U516 to be the culprit.
Anyway with a higher than normal TK of the resistors in the ADC hybrid, there is not that much one can do about it. For the 100K/100K pair one could in theory replace it with an external resistor pair - though not a very simple modification. For the 7V/10 V conversion it's more difficult to get a 70K/30K matched couple that is better than the original one. It's a question whether it is worth the effort.
As a crude workaround one could even go for an added temperature stabilization. Either a heater, or maybe a small fan.