I think that Aurelien Djament's answer is essentially correct, but I want to nitpick a bit.
If $\mathcal A$ is any locally finitely presentable category and $\mathcal C \subseteq \mathcal A$ is any strong generator of finitely-presentable objects, then every finitely-presentable object $X \in \mathcal A$ lies in the closure of $\mathcal C$ under finite colimits. So $X$ is a finite colimit of finite colimits of ... of finite colimits of objects of $\mathcal C$ -- an "$n$-fold" finite colimit of objects of $\mathcal C$. But $X$ need not be a "1-step" finite colimit of objects of $\mathcal C$. For example, I don't think every finitely-presented group is a finite colimit of copies of $\mathbb Z$.
One might strengthen the hypotheses and ask: if $\mathcal A$ is a locally finitely presentable category and $\mathcal C \subseteq \mathcal A$ is a dense generator, then is every finitely-presentable object $X \in \mathcal A$ a finite colimit of objects of $\mathcal C$? I don't know the answer to this.
But let's focus on the question at hand, i.e. the case where $\mathcal A = \hat {\mathcal C}$ is a presheaf category and $\mathcal C$ is the representables. Let $\tilde {\mathcal C}$ comprise the finite colimits of representables. Then indeed, $\tilde {\mathcal C}$ is closed under finite colimits. This is clear for finite coproducts -- just take the coproduct of the indexing diagrams for the colimits. Now let $A\rightrightarrows B \to C$ be a coequalizer where $A,B \in \tilde {\mathcal C}$. Then there is an epimorphism $\amalg_i X_i \to A$ and a coequalizer diagram $\amalg_j Y_j \rightrightarrows \amalg_k Z_k \to B$ where $X_i,Y_j,Z_k \in \mathcal C$ and the coproducts are finite. The composite maps $\amalg_i X_i \to A \rightrightarrows B$ lift to maps $\amalg_i X_i \rightrightarrows \amalg_k Z_k$. Then we have that $C$ is the coequalizer of the two induced maps $(\amalg_i X_i) \amalg (\amalg_j Y_j) \rightrightarrows \amalg_k Z_k$.
Now I claim that if $f,g \amalg_{i \in I} X_i \rightrightarrows \amalg_{k \in K} Z_k$ are two maps with coequalizer $C$, and if the $X_i$ are representable, then $C$ is the colimit of the following diagram. Indeed, for each $i \in I$, there is a unique $k = k_0(i) \in K$ such that $X_i \to \amalg_{i \in I} X_i \xrightarrow f \amalg_{k \in K} Z_k$ factors through $Z_k$, and similarly a $k_1(i)$ for $g$. The indexing set for our diagram has object set $I \amalg K$, and the nonidentity morphisms are a map $i \to k_0(i)$ and a map $i \to k_1(i)$ for each $i \in I$. Then $C$ is the colimit of the obvious diagram sending $i \mapsto X_i$ and $k \mapsto Z_k$. This diagram is finite if $I$ and $K$ are.
Thus in our case, $C \in \tilde{\mathcal C}$ as desired.
I want to emphasize that here we heavily used the fact that we're in a presheaf category.
I agree that any category which has finite colimits and filtered colimits has all colimits. But Aurelian's second bullet seems to suggest something stronger -- that if $X$ is a colimit of objects of $\mathcal C$, then $X$ is a filtered colimit of finite colimits of objects of $\mathcal C$. I don't have a counterexample, but I'm not sure this is true. The closest I can convince myself of is that $X$ is a coequalizer of coproducts of objects of $\mathcal C$, and therefore a coequalizer of filtered colimits of finite coproducts of objects of $\mathcal C$ -- but this only ensures that $X$ is a finite colimit of filtered colimits of finite colimits of objects of $\mathcal C$.
But using (3), Aurelian's third bullet goes through with some modification. As in any locally finitely presentable category $\mathcal A$ with strong generator $\mathcal C$, any finitely-presentable object is in the closure of the $\mathcal C$ under finite colimits. By (3), in the case $\mathcal A = \hat{\mathcal C}$, the closure of $\mathcal C$ under finite colimits consists exactly of $\tilde{\mathcal C}$, the objects which are "1-step" finite colimits of representables. Here, (3) is actually used in 2 places: first to ensure that the category $\tilde C \downarrow X$ is filtered (this being the diagram which indexes the canonical colimit for $X$), and second to ensure that $\tilde{\mathcal C}$ is closed under retracts.