What's an example of a locally presentable category "in nature" that's not $\aleph_0$-locally presentable?

The category of Banach spaces and contractions (over the reals or any other complete normed field, I think) is an example of an $\aleph_{1}$-presentable category which is not $\aleph_{0}$-presentable. The point is that the ground field is a strong generator and its represented functor $Ban(k,-)$ commutes with $\aleph_{1}$-filtered colimits. It essentially boils down to the fact about infinitary operations that arsmath pointed out in the above remark, details can be found in Borceux, Handbook of categorical algebra, vol. 2, Example 5.2.2 (e).

To see that the "unit ball functor" $Ban(k,-)$ does not commute with ordinary filtered colimits, you can take for example the identity $\varinjlim_{n < \omega} \ell^{1}(n) \cong \ell^{1}(\omega)$, where the $n$ are the finite ordinals and the maps the obvious inclusions. The set $\lim_{n < \omega} Ban(k,\ell^{1}(n))$ only consists of sequences with finitely many non-zero entries, while the set $Ban(k,\ell^{1}(\omega))$ has all summable sequences of norm $\leq 1$.

Here are examples appearing in algebraic topology.

The category of $\mathrm{Ext}$-$p$-complete abelian groups, as discussed in Homotopy limits, completions and localizations (Chapter VI, Sections 2-4) is locally $\aleph_1$-presentable but not locally $\aleph_0$-presentable. This follows from the fact that the $p$-adic integers $\mathbb{Z}_p$ are an object that is $\aleph_1$-presentable but not $\aleph_0$-presentable in that category.

Likewise, the category of $L$-complete modules, as discussed in Morava $K$-theories and localisation (Appendix A) is locally $\aleph_1$-presentable but not locally $\aleph_0$-presentable. More details can be found in this preprint (Appendix A.2).

The category of $\Delta$-generated spaces (the full category of the category of general topological spaces which are colimits of simplices) is locally $\lambda$-presentable for a regular cardinal $\lambda \geq 2^{\aleph_0}$. See http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html. This category contains all realizations of simplicial sets, all CW-complexes, all spheres, etc... i.e. all topological spaces used in algebraic topology. It is moreover cartesian closed.