I’ll show that $\Delta \to \mathbf{TOrd}$ preserves colimits. Let $F: J \to \Delta$ be a diagram that admits colimit $L$ with morphisms $\phi_j: F(j) \to L$ for each $j \in J$. I claim that $(L, \{\phi_j\}_{j \in J})$ is also the colimit of $F$ regarded as a functor to $\mathbf{TOrd}$. To that end, let $T$ be a totally ordered set and $\psi_j: F(j) \to T$ be a monotone map for each $j \in J$, s.t. $\psi_{j_2} \circ F(f) = \psi_{j_1}$ whenever $f: j_1 \to j_2$ is a morphism in $J$. For each nonempty finite subset $\tau = \{t_1, \cdots, t_n\}$ of $T$, where $t_1 < \cdots < t_n$, we define the collapsing morphism $c_\tau: T \to \tau$ by,
$$c_\tau(t) = \begin{cases}
t_n &, \, \mathrm{if} \, t \geq t_n\\
t_i &, \, \mathrm{if} \, t_i \leq t < t_{i+1} \, \mathrm{for} \, 1 \leq i < n\\
t_1 &, \, \mathrm{if} \, t < t_1
\end{cases}$$
Since $\tau \in \Delta$, there exists a unique morphism $u_\tau: L \to \tau$ s.t. $u_\tau \circ \phi_j = c_\tau \circ \psi_j$ for each $j \in J$.
Now, I claim that $\cup_{j \in J} \mathrm{range}(\psi_j)$ is finite. In fact, its cardinality is bounded by $|L|$. Assume to the contrary, then we may let $\tau \subset \cup_{j \in J} \mathrm{range}(\psi_j)$ be a finite subset of cardinality larger than $|L|$. But then by definition of $c_\tau$, we see that $\cup_{j \in J} \mathrm{range}(c_\tau \circ \psi_j)$ contains $\tau$. As $u_\tau \circ \phi_j = c_\tau \circ \psi_j$, this implies $\tau \subset \mathrm{range}(u_\tau)$. But $\tau$ has cardinality larger than $|L|$ whereas $u_\tau$ has domain $L$, which is a contradiction.
Now, let $\tau = \cup_{j \in J} \mathrm{range}(\psi_j)$, which is finite. Then regarding $u_\tau$ as a map with codomain $T$, we easily see that $u_\tau \circ \phi_j = \psi_j$ for all $j \in J$. Assume another map $u: L \to T$ has the same property. We observe that $\cup_{j \in J} \mathrm{range}(\phi_j) = L$. Indeed, we easily see that $\cup_{j \in J} \mathrm{range}(\phi_j)$, together with $\{\phi_j\}_{j \in J}$, is a colimit of $F$ in $\Delta$ by using similar methods as the collapsing morphisms defined above. So, the uniqueness of colimits ensures that $\cup_{j \in J} \mathrm{range}(\phi_j) = L$. But this means,
$$\tau = \cup_{j \in J} \mathrm{range}(\psi_j) = \cup_{j \in J} \mathrm{range}(u \circ \phi_j) = \mathrm{range}(u)$$
Now, by the property of colimits, and again because $\tau \in \Delta$, we see that $c_\tau \circ u_\tau = c_\tau \circ u$. But both $u_\tau$ and $u$ have ranges in $\tau$, so this implies $u_\tau = u$, concluding the proof that $L$ is the colimit of $F$ in $\mathbf{TOrd}$.
