Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$, $\Omega$ be an open subset of $M$, $\phi$ be a $C^1$-diffeomorphism$^1$ from $\Omega$ onto an open subset of $\mathbb R^k$, $U:=\phi(\Omega)$ and $\psi:=\phi^{-1}$.
Can we show that $\psi$ is an immersion, i.e. that $\psi$ is continuously differentiable and ${\rm D}\psi(u)$ is injective for all $u\in U$?
$\psi$ is clearly a topological embedding of $U$ into $M$. By assumption, there are an $\mathbb R^d$-open neighborhood $\tilde\Omega$ of $\Omega$, an $\mathbb R^k$-open neighborhood $\tilde U$ of $U$, $f\in C^1(\tilde\Omega,\mathbb R^k)$ with $\phi=\left.f\right|_\Omega$ and $g\in C^1(\tilde U,\mathbb R^d)$ with $\phi^{-1}=\left.g\right|_U$. Since $U$ is $\mathbb R^k$-open, it trivially follows that $\psi\in C^1(U,\mathbb R^d)$.
Turning to the injectivity claim, we may write $$\operatorname{id}_{\mathbb R^k}={\rm D}\operatorname{id}_U(u)={\rm D}(\phi\circ\psi)(u)\tag1$$ for all $u\in U$, but I'm not sure if we are really allowed to apply the chain rule now.
$^1$ If $E_i$ is a $\mathbb R$-Banach space and $B_i\subseteq E_1$, then $f:B_1\to E_2$ is called $C^1$-differentiable if $f=\left.\tilde f\right|_{B_1}$ for some $E_1$-open neighborhood $\Omega_1$ of $B_1$ and some $\tilde f\in C^1(\Omega_1,E_2)$ and $g:B_1\to B_2$ is called $C^1$-diffeomorphism if $g$ is a homeomorphism from $B_1$ onto $B_2$ and $g$ and $g^{-1}$ are $C^1$-differentiable.