Was one of Hilbert questions regarding physics to make an axiomatic foundation for physics? Regardless of Godels work could some Physics principles that are 'basic' and 'presently verifiable' be treated as axioms in a logic system along with Math axioms and Set theory or maybe with Category theory as a basis. If 'axiomatized' one could prove a physics theory with a type of Physics 'proving' algorithm. Like Leibnitz's dream of a 'calculating machine' that could calculate any question's answer.
Yes, it was. See also physics.stackexchange.com/q/87239/2451
Hilbert's approach is not really the same as physicists searching for the Theory of Everything. Physicists are concerned about truth, but not so much about the logical relations between one truth and another. Admittedly, a Theory of Everything would try to derive all physical laws from a very few ones, but Hilbert was explicitly interested in more than just deriving true laws from a few axioms. (For one thing, definitions are just as important as proofs. Now, physicists are often good enough at proofs, but rarely good at logical definitions. Nor should discovery be made to wait until the logic is all clear. Statistical Mechanics made great strides even though it suffered from logical contradictions and meaningless statements. They were all cleared up later.
Hilbert was interested in false physical theories as well as in true ones, for the sake of the light they might shed on the logic of true physical theories. He did something parallel to this in his researches on geometry: he would investigate a non-Archimedean geometry, which is obviously false, in order to show that the Archimedean axiom was necessary, could not be proved from the other axioms. He asked that the necessity of physical axioms be inveestigated in this way.
Hilbert influenced the axiomatisations of Quantum Mechanics by von Neumann, and although he had no influence on Dirac's axiomatisation, the two axiomatisations are really very similar except in some details.
But these axioms for Quantum Mechanics are unsatisfactory from a Hilbertian viewpoint. Wigner understood this viewpoint from his time in Hilbert's circle, and explained what was unsatisfactory: the same physical situation, the interaction of a particle with a measurement apparatus, can be analysed using the wave-function unitary time evolution axioms of QM, and you get one result: the future state of the particle + apparatus system is deterministically determined, but they are entangled. But if you analyse the same physical situation with the other axioms, the axioms about observables and the reduction of the wave packet, you get a probabilistic answer and one in which the measurement apparatus is treated as if it were classical, in a definite state, and not entangled. And the axioms do not tell you how these two results are related. There is no definition of measurement, event, or probability. No definition of "macroscopic state of the measurement apparatus". J.S. Bell also analysed these logical defects.
In 1900 Hilbert did not yet know about QM, but he was already asking for clarification of the logical status of probability in the relation between the micro-description of a gas via the Classical Mechanics of the individual molecules, and the macro-description of the gas as a whole in terms of temperature, the probability distribution of velocities of the molecules given by the Maxwell distribution which is parametrised by the temperature. He asked for a clarification of when micro-laws determine macro-behaviour, and how sensitive the macro-behaviour is to the exact form of the micro-laws. He also wanted a converse investigation: from the macro-behaviour, how much could you infer about the micro-laws? The basic problems for Classical Statistical Mechanics were all cleared up, logically, by Darwin and Fowler in the 1920's. Khintchine then popularised their logical framework in his book, Mathematical Foundations of Statistical Mechanics, Moscow, 1943.
It turns out that this is highly relevant to quantum measurement, as well. What one means physically by "macro-state" and "probability" has to be rigorously defined, and then the actual physics of the measurement process has to be analysed to see how it generates these results. Much progress has been made, see the cited link above.
It is not clear whether advances in our understanding of the logical nature and logical defnitions of "physical probability", "measurement", "macroscopic" and so on will lead to interesting Physics. But Hilbert did not think physicists would be either interested in or capable of investigating this Hilbert Problem, Hilbert's Sixth Problem. But mathematicians who have no physical intuition or do not know how Physics is done are not competent to contribute, either. Hilbert himself always hired a physics postdoc to explain to him the latest results, and his seminar often invited physicists to give talks (Einstein, for example).