Actually I successfully used D800E to shoot BIF, was not aware of response time differences
The shutter delay in viewfinder photography is a bit longer in the D810 because the mirror slows down before hitting the end of its travel, reducing the sound and vibration that it creates. I think the delay was just perceptible when first going from the D800 to D810 but I doubt it impacts my photography in any practical way. I recall when I got a used F-801s, I noticed the pronounced shutter delay, I don't recall the details but I think it was 250ms and was affected by whether automatic exposure was used or not and whether the exposure meter is active (by half-pressing the shutter before going all the way). Here we are talking about a difference of the order of 10ms.
This is an interesting report on the D5
http://www.imaging-resource.com/PRODS/nikon-d5/nikon-d5A6.HTMNikon has been able to slightly reduce the shutter lag with prefocused lens in viewfinder photography (39ms vs 43ms, D5 vs. D4s), much more significant differences were found when autofocus were used (132ms vs. 204ms; 60 AF-S Micro used as lens) or in live view mode (95ms vs 239ms). I think the 10ms difference between D810 and D800 is something easily accommodated for in my photography. In live view, the D810 has less shutter lag than the D800 (158ms vs. 184ms) and the delay between shots before is much shorter in the newer camera when using LV. So overall there is some improvement and Nikon live view is getting better.