Since ##U## is a space and spin rotation, it would be
$$U(R) = e^{-i\textbf{L}\cdot \hat{\textbf{n}}\phi/\hbar}\cdot e^{-i\textbf{S}\cdot \hat{\textbf{n}}\phi/\hbar}$$
And, then
$$\psi'(\textbf{r}, m) = \langle\textbf{r}, m|e^{-i\phi(\textbf{L} + \textbf{S}) \cdot...