It's very hard, if not impossible, to write such a book. Gödel's theorems are about the foundations of math, and so they don't actually use any significant mathematical results other than what they introduce. The main ideas, such as Gödel numbering, are essentially self-contained, and you can explain them to an audience of mathematically-minded people with enough patience.
On the other hand the proof of Fermat's Last Theorem depends crucially on various huge machineries developed throughout the 20th century, which are in turn based on and motivated by ideas from the 19th century. The new ideas involved all occur in a pre-existing conceptually advanced part of the mathematical spectrum, almost the polar opposite of logic and foundations in its reliance on earlier work.
One can try to give some ideas, but unfortunately it becomes extremely vague and confusing very fast. If the audience doesn't know what an elliptic curve is, one may be tempted to to speak vaguely of "doughnut shapes". For representation theory of Galois groups one could think of 'possible manifestations of the symmetry inherent in numbers', and instead of modular forms you might say 'highly symmetric functions'. But then you end up talking nonsense:
The proof involves showing that doughnut shapes are in some sense the same as highly symmetric functions. It had been shown beforehand that every doughnut surface has a highly symmetric function associated to it, but that this might be a bijection was a relatively new idea. It was established by recasting the problem in the world of possible manifestations of the symmetry inherent in numbers. Each doughnut shape results in such a manifestation, as does each highly symmetric function. One can then reformulate the bijection as a relation between the manifestations of symmetries of numbers associated to doughtnut shapes and to highly symmetric functions. Now one key idea is to consider these objects in a number system where numbers that have the same residue modulo $p$ are identified. It had been conjectured that certain manifestations of number symmetry in this world actually come from the regular world of manifestations of ...
It starts to sound like complete non-sense very quickly, and it would have to go on like that for dozens and dozens of pages.
On the other hand if you try to actually define what these things are and proceed rigorously you will not get very far. The Galois group is typically defined at the end of a year-long abstract algebra course. Even if you're willing to just consider examples and definitions, with no proofs for any theorems, you still need to define what a group is, what fields are, what field extensions are, what is a representation, etc. etc.
It's probably more fruitful to read about the general ideas involved instead, as opposed to how they were actually used. If a book can give a public audience some idea of what elliptic curves and modular forms are, that's already a very impressive accomplishment.
Another way to approach it is to read about the history of the problem, the people involved, and slowly the various ideas they introduced to tackle it. This would itself be a good introduction to modern number theory.
I like the Nova documentary 'The Proof'. It's fit for a public audience and does give them some ideas. I doubt you can expect much more, but I'd love to be proved wrong.