Regarding analyticity, there is the following nice result by Gomilko and Tomilov ([On Subordination of Holomorphic Semigroups][1], Theorem 1.1):

<cite authors="Gomilko, Alexander; Tomilov, Yuri">_Gomilko, Alexander; Tomilov, Yuri_, [**On subordination of holomorphic semigroups**](http://dx.doi.org/10.1016/j.aim.2015.05.016), Adv. Math. 283, 155-194 (2015). [ZBL1319.47034](https://zbmath.org/?q=an:1319.47034).</cite>

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.

  [1]: https://arxiv.org/pdf/1408.1417.pdf