This is a standard fact about the inverse method.
For $n=0,1,\ldots$ and $u\in(0,1)$, let $$G_n(u):=\sup\:\{x\in\mathbb R:F_n(x)\le u\}.$$
It is easy to check that $G_n$ is non-decreasing (and right-continuous*). In particular $G_0$ (just like $F_0$) has at most countably many point of discontinuities.
We now show that $G_n(u)\to G(u)$ for all $u\in(0,1)$ such that $G_0$ is continuous at $u$.
- Let $0<\varepsilon<u$. Let $x\in\mathbb R$ be such that $F_0(x)\le u-\varepsilon$. Up to choosing an arbitrarily closer $x$, we can assume that $F_0$ is continuous at $x$. Then the weak convergence of distributions gives $F_n(x)\to F_0(x)$ as $n\to\infty$. In particular $F_n(x)\le F_0(x)+\varepsilon$ for all $n$ large enough.
This implies that $x\le G_n(F_0(x)+\varepsilon)$ for such $n$, and then (because $F_0(x)\le u-\varepsilon$), $x\le G_n(u)$. Thus $$\liminf_{n\to\infty}G_n(u)\ge x$$
for any chosen $x$ with $F_0(x)\le u-\varepsilon$, so $$\liminf_{n\to\infty}G_n(u)\ge G_0(u-\varepsilon)$$
by definition of $G_0(u-\varepsilon)$. As $\varepsilon$ was arbitrary and $G_0$ is assumed to be continuous at $u$, we deduce that $$\liminf_{n\to\infty}G_n(u)\ge G_0(u).$$
- For the other direction, let $x>G_0(u)$. Then $F_0(x)>u$ and again, the convergence in distribution gives $F_n(x)>u$ for all $n$ large enough, so $G_n(u)\le x$. Hence $$\limsup_{n\to\infty}G_n(u)\le x.$$ This holds for all $x>G_0(u)$ so $$\limsup_{n\to\infty}G_n(u)\le G_0(u).$$
Therefore $G_n(u)\to G_0(u)$ for all point $u\in(0,1)$ of continuity of $G_0$. As a uniform variable $U\in(0,1)$ is almost surely a continuity point of $G_0$, we get that $G_n(U)\to G_0(U)$ almost surely (and obviously also in probability).
*: We also encounter (perhaps more often) the left-continuous inverses $\inf\:\{x\in\mathbb R:F_n(x)\ge u\}$ for which the above fact can be proved similarly.