Hilbert reconstructed Euclid's axioms. Is there an equivalent restructuring of Newton's axioms, or are they considered consistent?
Would you consider clarifying the question itself to enable a more informative answer than this? Particularly, can you be more specific about what you mean by 'restructuring', in a sense that could equally apply to alterations in Euclid's axioms and/or in Newton's laws of motion?
In all three editions of Newton's 'Principia' the axioms that occur near the beginning are not just 'axioms', they are 'axioms or laws of motion'.
They have an essentially physical aspect that means they can't meaningfully be treated as purely mathematical constructs that might be differently chosen, they were selected for their compatibility with natural phenomena of motion and related physical behavior: and any alteration of them that is contemplated must clearly maintain that compatibility.
Having said that, the 'axioms or laws of motion' have often been re-expressed. They are usually expressed nowadays, especially in modern physics textbooks, in a way that looks very different indeed than their original expression by Newton. Yet it is widely considered that the differences are or ought to be of modernised form and expression only, and not of substance or core meaning. Yet arguments can arise over such questions as whether the modern formulations really do match in substance what Newton meant, and sometimes even what did he mean?
I will try to go further if you would clarify the question.