Bumping an old question, I believe that Andreas' bound of the next admissible ordinal is in general (and I suspect always) **not sharp**. The crucial point is not the theory of the model in question, but rather its closure properties - in particular, the bound of the next admissible fails to be sharp whenever $M^\omega\subseteq M$. *(It's possible of course that large cardinals will be relevant in case $M^\omega\subseteq M$, but I actually think the failure of sharpness will hold unconditionally.)*


**Part of the point of this answer is that the notation in the OP winds up being highly misleading.** So let me notationally start from scratch:

 - I'll write "$\Delta(M)$" for the supremum of the definable-over-$M$ well-orderings of subsets of $M$. 

 - For $M$ a transitive model of KP + Infinity (= KP$\omega$), I'll write "$Ad(M)$" for the smallest transitive model of KP$\omega$ with $M$ as an element.

 - I'll write "$\omega_1^{CK}(M)$" for $Ad(M)\cap Ord$.

****

(*[This MSE answer of mine](https://math.stackexchange.com/a/3471324/28111) contains the argument below in more detail, and [this old MO answer of mine](https://mathoverflow.net/a/277345/8133) is also relevant:*)

The key observation is that if $M^\omega\subseteq M$ then any $M$-definable ordering of a subset of $M$ which is ill-founded has a descending sequence in $M$. This means that the set $\mathcal{W}_M$ of formulas-with-parameters-in-$M$ which define in $M$ well-orderings of subsets of $M$ is **computable in $Ad(M)$**: this is because, given a candidate $\varphi$, we can just search in $Ad(M)$ for either an isomorphism between $\varphi^M$ and some ordinal or a descending sequence through $\varphi^M$ *(in fact, all we're really using here is $M^\omega\subseteq Ad(M)$)*. This means that $$\Delta(M)<\omega_1^{CK}(M)$$ whenever $M$ is sufficiently closed.

As I said above, I strongly suspect that this upper bound is *never* sharp, but I don't see an argument for that at present. Annoyingly, there seems to be a paucity of candidates for better bounds. The height $\eta_M$ (my notation) of the smallest [$E$-closed set](https://projecteuclid.org/euclid.pl/1235422642) containing $M$ as an element is a possible candidate, being in general *(even always, I think)* smaller than $Ad(M)$ when $M$ is admissible, but the plausibility of $\eta_M=\Delta(M)$ is due entirely to my ignorance of $E$-recursion.