Two different camera modules will have two different white balance results. However, I would expect them to be close to the same.
Check the LR presets you are using to see if they are using a "auto" white balance as this will cause LR to search each photo for something it thinks is supposed to be white or neutral gray.
"As shot" might be the setting you are looking for if not using a "daylight" or numeric setting.
I have a few User presets I created, they are simple adjustments with "As shot" WB. I rarely if ever use "Auto". When I start working on images from a shoot, I import, apply a basic Dev, leave WB at "As shot". Then I'll go to an image from each camera and set WB accordingly.
But for this question I'm posing, the WB between the two cameras looks different. If I take an image from both bodies, side by side in comparison, they are of the same subject taken seconds apart, using the exact same basic LR Dev settings, then with temp set at 5000 and tint set at 0, the WB results are different.
I would expect two different white balance results from two different camera models. Algorithms get tweaked as new models come out. The results should be similar, meaning close to being the same.
Checking the camera manuals to see what the various WB settings mean between the two models, The D800 and D810 use the same temperatures for the WB presets. However, if you have both cameras set to "auto" WB, expecting the two results to be identical would not be a reasonable expectation.
In your example photos, how much different are we talking about for the WB results?
The difference: is the D810 is about 1000 cooler on temp, and about +5-6 more purple on the tint. Meaning, I need to adjust the D810 up 1000 in temp, and move down 5-6 in tint - when compared to D800. Or, in LR the D800 looks overall warmer and shifted more to the green, while the D810 in LR looks overall cooler and shifter more to the purple.
Both bodies are set to Auto1. But, is that even relative when shooting in RAW? It's a B&W negative, with no color profile. LR is applying a profile for the color rendering, correct?
The Nikon D800 and D810 use the same exact sensor, but the low-pass filter (OLPF) is removed on the D810. Technically the OLPF may affect the White Balance, but I would assume Nikon to have corrected that in-camera. At any rate I wouldn't expect to see a 1000K shift from removing the OLPF.
Try a test using a white balance card or even a piece of poster board. Shoot the card with both cameras using the same lens and settings. Make sure the card has the exact same lighting and fills the frame. Inside LR use the WB Eydropper on the same area of the card and post the WB readings here. This is just to rule out the lens or some other scene related issue as the cause. If you still see a 1000K WB difference then I suggest contacting Nikon Customer Service.
1000 degrees is a big difference.
How do the photos looks if you are using Nikon software to view them?
Actually the sensors are different, although I don't know exactly how different. The base ISO in the D800 is 100, and in the D810 it's 64.
The removed low pass filter (or rather cancelled out) was optional as the D800E. The new model offers no filtered option at all.
I'd expect a slight difference in WB, but I agree 1000K seems a lot.