$\newcommand\ep\varepsilon$Without loss of generality, $\ep\in(0,1/2]$. Let $$A:=A_0\cup A_1\cup F,$$ where $A_0:=[0,\ep)\times B$, $A_1:=(1-\ep,1]\times B$, $F:=[0,1]\times\{0\}$, and $B$ is the unit ball.
You have a well-defined function $f$ on $A$ such that $f|_{A_0}=\rho_0$, $f|_{A_1}=\rho_1$, and $f(\cdot,0)=F_0$.
Your question is then when $f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.
For that it is clearly necessary that $\rho_0$ and $\rho_1$ be extendible to smooth functions on the closures $\bar A_0$ and $\bar A_1$ of $A_0$ and $A_1$. Then $f$ can be accordingly extended to a uniquely determined function $\bar f$ on the closure $\bar A$ of $A$.
So, the question then becomes when $\bar f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$.
By Whitney's Theorem I, a sufficient condition for this is that $\bar f$ be of class $C^\infty$ on $\bar A$ in the sense of this paper by Whitney (the set $A$ in the mentioned theorem can be any closed subset of a Euclidean space -- it does not have to be a manifold).
It is clear that this sufficient condition is also necessary.
Therefore and because the functions $\rho_0$, $\rho_1$, and $F_0$ are smooth, we conclude:
$f$ can be extended to a smooth function $F$ on $C=[0,1]\times B$ iff condition (3.2) in the mentioned paper by Whitney holds for the function $\bar f$ and the points $x^0=(\ep,0)$ and $x^0=(1-\ep,0)$ in the set $\bar A$ (with respect to the set $\bar A$).