EDIT (12/18/15)
The below argument using countably-generated spaces achieves the estimate that $Delta$-generated spaces are locally $(2^{2^{aleph_0}})^+$-presentable. An improvement (probably optimal in light of Zhen Lin's comment) to local $(2^{aleph_0})^+$-presentability can be obtained by using sequential spaces instead of contably-generated ones, and an axiomatization of sequential spaces by Gutierres and Hoffman. The same ideas are applicable: by axiomatizing spaces in terms of convergence, it becomes easy to compute sufficiently-filtered colimits.
I'll keep the below argument here, though, because it easily generalizes to show that the $mathcal{A}$-generated spaces are locally presentable for any small full subcategory $mathcal{A} subset mathsf{Top}$.
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.
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