Consider the flow on $\Omega={\mathbb R}^2$ given by $$\phi_t(x,y)=(e^tx, e^{-t}y).$$
Take $U_1$ equal to the open upper half-plane ($y>0$) and $U_2$ equal to the open right half-plane  ($x>0$). I leave it to you to verify that the flow is proper on both $U_1, U_2$ and is not proper on their union.