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I know the question is old, but is possible to give an exact characterization to $d$, at least if it's a prime number:

It is known that considering a prime $p \equiv 1\mod 4$, there is always a solution to $x^2 - py^2 = -1$ in integers, the proof is from Mordell "Diophantine Equations" pages 55-56 (thanks to @Will Jagy for pointing that out).

Furthermore Dirichlet proved that if the prime $p \equiv 1 \mod 4$ and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$ (the latter with $2^{(p-1)/4}\equiv-1\bmod p$), the equation $x^2 - 2py^2 = -1$ has still integer solutions (reference: https://www.forgottenbooks.com/en/download/ThePellEquation_10024828.pdf pag. 80)

So you can say that if $p$ is a prime and $p \equiv 5 \mod 8$, or if $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, both NPEₚ NPE₂ₚ have integer solutions

I know the question is old, but is possible to give an exact characterization to $d$, at least if it's a prime number:

It is known that considering a prime $p \equiv 1\mod 4$, there is always a solution to $x^2 - py^2 = -1$ in integers, the proof is from Mordell "Diophantine Equations" pages 55-56 (thanks to @Will Jagy for pointing that out).

Furthermore Dirichlet proved that if the prime $p \equiv 1 \mod 4$ and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$ (the latter with $2^{(p-1)/4}\equiv-1\bmod p$), the equation $x^2 - 2py^2 = -1$ has still integer solutions (reference: https://www.forgottenbooks.com/en/download/ThePellEquation_10024828.pdf pag. 80)

So you can say that if $p$ is a prime and $p \equiv 5 \mod 8$, or if $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, both $\text{NPE}_d $ and $ \text{NPE}_{2d} $ have integer solutions.

Update

I noticed that if we represent $ 2p = a^2 + b^2 $, if it holds that $a\equiv\pm 3\bmod 8$ and $ b\equiv\pm 3\bmod 8 $, then $ x^2 - 2py^2 = -1 $ is not solvable, but I don't know how to prove it. This maybe is related with the GLW theorem that the OP highlithed

I know the question is old, but is possible to give an exact characterization to $d$, at least if it's a prime number:

It is known that considering a prime $p \equiv 1\mod 4$, there is always a solution to $x^2 - py^2 = -1$ in integers, the proof is from Mordell "Diophantine Equations" pages 55-56 (thanks to @Will Jagy for pointing that out).

Furthermore Dirichlet proved that if the prime $p \equiv 1 \mod 4$ and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, the equation $x^2 - 2py^2 = -1$ has still integer solutions (reference: https://www.forgottenbooks.com/en/download/ThePellEquation_10024828.pdf pag. 80)

So you can say that if $p$ is a prime and $p \equiv 5 \mod 8$, or if $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, both NPEₚ NPE₂ₚ have integer solutions

I know the question is old, but is possible to give an exact characterization to $d$, at least if it's a prime number:

It is known that considering a prime $p \equiv 1\mod 4$, there is always a solution to $x^2 - py^2 = -1$ in integers, the proof is from Mordell "Diophantine Equations" pages 55-56 (thanks to @Will Jagy for pointing that out).

Furthermore Dirichlet proved that if the prime $p \equiv 1 \mod 4$ and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$ (the latter with $2^{(p-1)/4}\equiv-1\bmod p$), the equation $x^2 - 2py^2 = -1$ has still integer solutions (reference: https://www.forgottenbooks.com/en/download/ThePellEquation_10024828.pdf pag. 80)

So you can say that if $p$ is a prime and $p \equiv 5 \mod 8$, or if $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, both NPEₚ NPE₂ₚ have integer solutions

I know the question is old, but is possible to give an exact characterization to $d$, at least if it's a prime number:

It is known that considering a prime $p \equiv 1\mod 4$, there is always a solution to $x^2 - py^2 = -1$ in integers, the proof is from Mordell "Diophantine Equations" pages 55-56 (thanks to @Will Jagy for pointing that out).

Furthermore Dirichlet proved that if the prime $p \equiv 1 \mod 4$ and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$, the equation $x^2 - 2py^2 = -1$ has still integer solutions (reference: https://www.forgottenbooks.com/en/download/ThePellEquation_10024828.pdf pag. 80)

So you can say that if $p$ is a prime and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$ both NPEₚ NPE₂ₚ have integer solutions

I know the question is old, but is possible to give an exact characterization to $d$, at least if it's a prime number:

It is known that considering a prime $p \equiv 1\mod 4$, there is always a solution to $x^2 - py^2 = -1$ in integers, the proof is from Mordell "Diophantine Equations" pages 55-56 (thanks to @Will Jagy for pointing that out).

Furthermore Dirichlet proved that if the prime $p \equiv 1 \mod 4$ and $p \equiv 5 \mod 8$ or $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, the equation $x^2 - 2py^2 = -1$ has still integer solutions (reference: https://www.forgottenbooks.com/en/download/ThePellEquation_10024828.pdf pag. 80)

So you can say that if $p$ is a prime and $p \equiv 5 \mod 8$, or if $p \equiv 9 \mod 16$ with $2^{(p-1)/4}\equiv-1\bmod p$, both NPEₚ NPE₂ₚ have integer solutions