How do the units work for the derivation of $\dot{\hat{r}}$ and $\dot{\hat{\phi}}$? Isn't $\Delta \phi$ in radians and $\Delta r$ in meters, so units are inconsistent?

Angles are considered ``dimensionless'', and all the terms in the $\dot{\vec{r}}$ expression have dimensions of length per time (e.g., meters per second) and all the terms in the $\ddot{\vec{r}}$ expression have dimensions of length per time squared (e.g., meters/s$^2$).