I give an answer.

The condition $F\in C^1$ guarantees the uniqueness of solutions to the ODE system, namely, the characteristic curves. Using this, one can show that the map $x(s,\sigma)$ is $C^1$ in the $s$-variable, by differentiable dependence of the initial data (I don't remember where I see that, but I'm pretty sure of this) and $C^2$ in the $\sigma$-variable by construction. So, the inverse function theorem applies and one can conclude that $u\circ x^{-1}$ is a $C^1$ (local) solution to the problem.

If we suppose that $F \in C^0$ or $F\in C^{0,\alpha}$ we don't have uniqueness for the ODE system and we cannot have differentiable dependence of the initial data. Hence, in general, the map $x \not\in C^1$ and we cannot conclude.