Here's a sketch of a more direct proof that $\Delta$-generated spaces (call this category $\Delta-\mathrm{Gen}$) are locally presentable. It's essentially a "compiling-out" of Fajstrup and Rosický's proof. The best estimate I'm able to extract for the accessibility rank is $(2^{2^{\aleph_0}})^+$, though from Zhen Lin's comment one should probably expect the true accessibility rank to be $(2^{\aleph_0})^+$.

First we show that the category $\aleph_0-\mathrm{Gen}$ of countably generated spaces -- those spaces which are to the countable topological spaces as $\Delta$-generated spaces are to simplices -- is locally presentable. Since $\Delta-\mathrm{Gen}$ is a full subcategory of $\aleph_0-\mathrm{Gen}$ which is closed under colimits, with a dense generator given by the simplices, it is also locally presentable.

(That argument might sound like it requires Vopenka's principle, but it doesn't: it just uses the characterization of locally presentable categories as those cocomplete categories with a dense generator [this hypothesis can be weakened to: strong generator] of presentable objects. If $\mathcal{K}$ is locally presentable then every object there is a cardinal $\lambda$ such that the object is $\lambda$-presentable [since for every object there is a $\lambda$ such that it is a $\lambda$-small colimit of canonical generators, and the $\lambda$-presentable objects are closed under $\lambda$-small colimits]. If $\mathcal{L}$ is a full subcategory closed under colimits, then all the objects of $\mathcal{L}$ are also presentable, so if $\mathcal{L}$ has a dense generator, it is locally presentable. Explicitly in this case, the simplices are continuum-sized colimits of countable spaces, so they are presentable. If the countable spaces were $(2^{\aleph_0})^+$-presentable, the simplices would be too; as it is though, I can only show that the countable spaces are $(2^{2^{\aleph_0}})^+$-presentable, so that's the best estimate I have for the simplices, too.)

The reason for bringing $\aleph_0-\mathrm{Gen}$ into the picture is that in $\aleph_0-\mathrm{Gen}$, it's easy to describe the topology on a colimit $X = \varinjlim X_i$ when the colimit is sufficiently filtered. Namely, $X$ has the topology where

a countably-supported ultrafilter $\mathcal{F} \in \beta_\omega X$
converges to a point $x \in X$ if and only if "$\mathcal{F}$ already
converges to $x$ at some stage of the colimit", i.e. iff there exists
an $X_i$ and an $x_i \in X_i$ mapping to $x$ and a countably-supported
ultrafilter $\mathcal{F}_i \in \beta_\omega X_i$ which pushes forward
to $\mathcal{F}$, such that $\mathcal{F}_i$ converges to $x_i$ in
$X_i$.

Here we use the notion of ultrafilter convergence: an ultrafilter $\mathcal{F} \in \beta X$ is said to *converge* to a point $x \in X$ iff every neighborhood of $x$ is an element of $\mathcal{F}$. A *countably-supported* ultrafilter $\mathcal{F} \in \beta_\omega X$ is just an ultrafilter which contains a countable subset of $X$.

It's obvious from this description of a sufficiently-filtered colimit that spaces of sufficiently small cardinality are presentable, because a function between (countably-generated) topological spaces is continuous iff it sends convergent (countably-supported) ultrafilters to convergent ultrafilters. Since the countable spaces are dense in $\aleph_0-\mathrm{Gen}$, it follows that $\aleph_0-\mathrm{Gen}$ is locally presentable.

The subtlety, of course, comes in verifying that this description of ultrafilter convergence in a sufficiently-filtered colimit actually arises from a topology (and that this topology is countably-generated). Barr showed that a relation $R \subseteq \beta X \times X$ defined for all ultrafilters arises from a topological space if and only if $R$ is a lax algebra for the ultrafilter monad $\beta$, giving a (concrete) equivalence of categories between topological spaces and lax $\beta$-algebras. By replacing $R$ with a relation $R \subseteq \beta_\omega X \times X$ in this definition, we get a "lax-algebraic" description of $\aleph_0-\mathrm{Gen}$, which we can use to compute sufficiently-filtered colimits as above.

To be precise about the ultrafilter description of $\aleph_0-\mathrm{Gen}$, let me first review the ultrafilter description of general topological space. Consider a relation $ \beta X \overset{\pi_1}{\leftarrow} R \overset{\pi_2}{\to} X$, and write $\mathcal{F} \rightsquigarrow x$ if $(\mathcal{F},x) \in R$, i.e. $R= \{(\mathcal{F},x) \mid \mathcal{F} \rightsquigarrow x\}$. Then $R$ is the convergence relation for a topology on $X$ if and only if the following conditions hold:

For every $x \in X$, $\mathrm{prin}(x) \rightsquigarrow x$, where $\mathrm{prin}(x)$ is the principal ultrafilter at $x$.

If $\mathcal{G}$ is an ultrafilter on the set $R$ itself, and if $(\pi_2)_*(\mathcal{G}) \rightsquigarrow x$, then $\sum (\pi_1)_*(\mathcal{G}) \rightsquigarrow x$.

Here $()_*$ is the pushforward of ultrafilters, $f_*(\mathcal{F}) = \{A \mid f^{-1}(A) \in \mathcal{F}\}$ and $\sum: \beta \beta X \to \beta X$ is the sum of ultrafilters $\sum \mathcal{H} = \{A \mid \hat{A} \in \mathcal{H}\}$, where $\hat{A} = \{\mathcal{F} \mid A \in \mathcal{F}\}$.

Analogously, consider a relation $ \beta_\omega X \overset{\pi_1}{\leftarrow} R \overset{\pi_2}{\to} X$, with the notation $\mathcal{F} \rightsquigarrow x$ as before. Then $R$ is the convergence relation (restricted to countably-supported ultrafilters) for a countably-generated topology on $X$ if and only if the following conditions hold:

For every $x \in X$, $\mathrm{prin}(x) \rightsquigarrow x$, where $\mathrm{prin}(x)$ is the principal ultrafilter at $x$.

If $\mathcal{G}$ is an ultrafilter on the set $R$ itself, and if $(\pi_2)_*(\mathcal{G}) \in \beta_\omega X$ and $(\pi_2)_*(\mathcal{G}) \rightsquigarrow x$, and if $\sum (\pi_1)_*(\mathcal{G}) \in \beta_\omega X$, then $\sum (\pi_1)_*(\mathcal{G}) \rightsquigarrow x$.

Note that in (2), $\mathcal{G}$ is not required to be countably supported, nor is $(\pi_1)_*(\mathcal{G})$. So a countably-generated space is not apparently the same thing as a lax algebra for the monad $\beta_\omega$ of countably-supported ultrafilters -- it needs to satisfy a stronger associativity condition (2) which refers back to the full ultrafilter monad $\beta$. There ought to be some general 2-categorical or equipment-theoretic description of the relationship between these two monads and of this sort of "hybrid" lax algebra for them, but I haven't worked out what it should be.

I doubt that $\Delta-\mathrm{Gen}$ can be described directly in terms of a submonad of the ultrafilter monad $\beta$ -- this is the reason for bringing countably-generated spaces into the story.