Has Fermat's Last Theorem per se been used?

Corollary 3.17 in this paper of Stefan Keil uses FLT for exponent 7 to show that if $E/\mathbb{Q}$ is an elliptic curve with a rational 7-torsion point $P$, and $E\rightarrow E'$ is the 7-isogeny with kernel $\langle P\rangle$, then $E'(\mathbb{Q})[7]=0$. There are of course lots of ways of proving this, but the paper does it by writing down a parametrisation of all elliptic curves over $\mathbb{Q}$ with 7-torsion and of their rational 7-isogenies, and then playing with parameters to get a contradiction to FLT.

It is perhaps an indication of the average age of today's MOers that nobody remembers the work of Hellegouarch who introduced around 1970 what is nowadays called the Frey curve precisely in order to deduce information about elliptic curves from (the then) known results about Fermat's Last Theorem.

Do applications to physics count?

Supersymmetry Breakings and Fermat's Last Theorem, Hitoshi Nishino, Mod.Phys.Lett. A 10 (1995) 149-158.

In this paper, we give the first application of Fermat's last theorem (FLT) to physical models, in particular to supersymmetric models in two or four dimensions. It is shown that FLT implies that supersymmetry is exact at some irrational number points in parameter space, while it is broken at all rational number points except for the origin. This mechanism presents a peculiar link between the FLT in number theory and the vacuum structure of supersymmetry. Previously, the only well-known connection between number theory and supersymmetry has been via topological effects, such as instantons and monopoles in supersymmetric models.