Can you try the same using Julia 1.11?
If you use juliaup switching between Julia version is trivial.
Installing juliaup is a simple as:
curl -fsSL https://install.julialang.org | sh
When you have it installed, you can do:
juliaup add 1.11
juliaup default 1.11
or
juliaup add 1.9
juliaup default 1.9
to switch back to your old version.
And did you try to install it into an empty project?
mkdir test
cd test
julia --project
and then in Julia:
using Pkg
pkg"add NeuralPDE"
?
Does that work?