For a D810, the Sigma 150-600mm Contemporary is very good at wildlife. You can buy much better, but not a whole lot better for the money invested.
My own Milky Way photos to date were done with a crop sensor D7000 and Sigma 18-35mm f/1.8 Art, then later a D750 with Sigma 20mm f/1.4 Art. The 18-35mm is a DX lens and really out of consideration for the D810. You want as wide of a lens as possible with as big of an aperture as can be had, all within your budget consideration.
Since the last time I did a Milky Way photo, I have purchased a barely-used Carl Zeiss 15mm f/2.8 Distagon and I am eager to try that lens. But advising others, I would say look at Nikon's 20mm f/1.8G lens. It is less than the Sigma Art lens but trying it at the store I felt like it is very comparable. I know that astrophotographers often use either the Samyang or Rokinon 14mm f/2.8 manual focus lenses. You are going manual mode anyhow with the Milky Way, and they are certainly a lot less expensive than other choices. My used Carl Zeiss lens came in about 2x the Samyang price (and that was an extraordinary low price) but I have been enjoying it as a daytime landscape lens also.
Full disclosure, I am low-level shopping for a Z-mount 14-24mm f/2.8 for my Z5 in the used market. When I finally land one, the Sigma 20mm f/1.4 Art lens likely will be sold off. I only have been using that lens for Milky Way. The Zeiss 15mm has already been used way more by me. I would have bought 1 last week if not for also getting a bill for a CT scan last month that was even more than the lens.
Some of my samples using D750 body.
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