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Given what is publicly known about the Mill CPU, would it be affected by Spectre(-NG) or Meltdown style attacks? Why/why not?

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Mill had published a sort of white paper on the extent of their vulnerability to Spectre and Meltdown. Since we don't know much about Spectre-NG yet (probably the first wave would come out on 7th May 2018), I am going to rely on my assumption that the new variants attack the Hypervisor and Intel's secure guard extension.

The attacks on hypervisor (using software level speculative execution) definitely can affect Mill's hardware or probably any hardware in general. Here is the relevant section from the whitepaper:

There is little that a vendor can do about victim code that deliberately loads from an attacker supplied address without checking the validity of the address first; such code is a bug, plain and simple. Exploitable speculation introduced by the compiler as an optimization is also a bug of course, but more insidious because inspection of the victim source will not reveal the problem.

The Mill is a very wide-issue machine, which allows the compiler to improve performance by carefully interleaving speculative execution using meta-data and predicated (conditional) instructions. For example, it is common for the two legs of an if-statement to be interleaved and both sides executed unconditionally but with one side masked out via meta-data or not used by subsequent conditional instructions.

As for attacks on Intel's secure guard extension:
According to the white paper, Mill architecture seems to be secure against such attacks.

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