Both Hawking and Dyson have said that Godel's incompleteness theorems prove that it is impossible for us to formulate an absolutely fundamental Theory of Everything. Is that true? Do the theorems apply to the physical world as they apply to the realm of mathematics?

Hawking has a habit of making claims in order to attract media attention. IMHO you can safely ignore him, his contribution to physics is scant, and whilst a media darling, he's not nearly so popular with professional physicists. See this physicsworld article: "Hawking explained that M-theory allows the existence of a “multiverse” of different universes, each with different values of the physical constants. We exist in our universe not by the grace of God, according to Hawking, but simply because the physics in this particular universe is just right for stars, planets and humans to form. There is just one tiny problem with all this – there is currently little experimental evidence to back up M-theory. In other words, a leading scientist is making a sweeping public statement on the existence of God based on his faith in an unsubstantiated theory." Dyson isn't the same, and you can see him saying some reasonable-sounding things in articles like this. But I've exchanged emails with him, and he came across as rather disparaging of Einstein's attempts to unify electromagnetism and gravity. Maybe he has a bit of emotional interest in saying we can't make scientific progress. Kind of not invented here, if you know what I mean. No. Energy is absolutely fundamental, and Godel's incompleteness theorem doesn't have anything to do with light and matter and electromagnetism and gravity and cosmology. No.

This attempt at a rebuttal has absolutely no connection to Godel's Incompleteness Theorems or what they actually say. What the incompleteness theorems actually show is that it's impossible to prove every statement that can be constructed in a consistent first-order logical system. If you have a set of basic axioms and some propositions which satisfy those axioms, it's always possible to make certain secondary statements about those propositions which themselves cannot be proven true or false, and such secondary statements must themselves be taken as axioms. For instance, in number theory you can lay out all the basic axioms which define a natural number, but no one has found a means of using those axioms to prove that the numbers which satisfy them also obey the law of induction. In fact, as I understand it, Godel specifically showed induction to be unprovable, by constructing a system of pseudo-numbers which obey all the same axioms but don't satisfy induction. There's nothing in logic forbidding physics from utilizing well-defined mathematics to describe the universe, regardless of whether all the postulates in those mathematics can be proven strictly from first-order axioms. However, Godel did find an unphysical solution to the equations of General Relativity in which closed timelike curves are possible (i.e. a particle encountering itself in the past), and this had Einstein convinced that the theory couldn't possibly be complete, since he felt it ought to self-consistently describe any situation which satisfies its equations, even if that situation is impossible to achieve in the real universe.

The moot point is this: nor does physics. A unified theory or ToE doesn't have to. Not a problem. And there's nothing that limits physics to mathematical axioms and formal proofs. In physics, we have scientific evidence. Einstein already knew the theory was incomplete. See for example this 1929 article where he said this: "It can, however, scarcely be imagined that empty space has conditions or states of two essentially different kinds, and it is natural to suspect that this only appears to be so because the structure of the physical continuum is not completely described by the Riemannian metric." Also see A World without Time, the Forgotten Legacy of Gödel and Einstein. The message delivered by this book is that Gödel convinced Einstein that "time does not exist as we ordinarily understand it" and that "time does not pass". And that the particle cannot encounter itself in the past because it doesn't move along a worldline. See this page for something about that: Please Register or Log in to view the hidden image!

If the "absolutely fundamental Theory of Everything" to which you refer derives of a system of mathematical reasoning, then the theory must be either INCOMPLETE or INCONSISTENT within the limits of that system of reasoning. Our minds, although finite, are not limited by all of the typical confines of a system of mathematical reasoning that is based on the use of symbols. The use of symbols (including any symbolic system of representation which may be vocal, written, or similarly encoded) are the tools of a finite mind. Unlike a rules based intelligence like a computer system, we do have means at our disposal for overcoming halting problems, not the least of which is to construct new systems of reasoning to solve new problems as required. All of the truth of everything will literally not fit within the available capacity of our minds, and this is why falsification of the false ideas we encounter rather than the affirmation of ideas that we understand to be true, are the lingua franca of our limited cognition. Even though Hawking and Dyson were referenced in the OP, this question properly belongs in a discussion of Philosophy, NOT Physics and Math.