Regarding analyticity, there is the following nice result by Gomilko and Tomilov (On Subordination of Holomorphic Semigroups, Theorem 1.1):
Gomilko, Alexander; Tomilov, Yuri, On subordination of holomorphic semigroups, Adv. Math. 283, 155-194 (2015). ZBL1319.47034.
A smooth function $\psi\colon (0,\infty)\to (0,\infty)$ is called Bernstein function if $(-1)^n f^{(n+1)}\geq 0$ for all $n\in\mathbb{N}$. If $\psi$ is a Bernstein function and $A$ generates a bounded holomorphic semigroup of angle $\theta$ on the Banach space $X$, then $\psi(A)$ also generates a bounded holomorphic semigroup of angle $\theta$.
Clearly, $\lambda\mapsto \lambda^s$ is a Bernstein function for $s\in (0,1)$. Thus the fractional Laplacian generates a bounded holomorphic semigroup whenever the Laplacian does. In particular, this is the case on $L^p(\Omega)$ for $p\in[1,\infty)$. As a consequence you get $W^{\alpha,p}$ regularity of $u(t,\cdot)$ in terms of $u_0$ and $f$ by the usual semigroup methods. Of course, if you are only interested in the case $p=2$, all of this can also be established via the spectral theorem.
