Thus the pushout functor $colim: Top^P \rightarrow Top$ does not take weak equivalences to weak equivalences and we can not invoke the universal property of $Ho(Top^P)$ to get a functor $Ho(colim):Ho(Top^P) \rightarrow Ho(Top)$ making the square commute which has $colim$ on top, $Ho(colim)$ at the bottom and the projections to the homotopy categories at the left and right. The best we can do is to find the initialterminal functor "$hocolim$" which allows filling this square with a natural transformation - which is exactly the Kan extension property you cited.
where the middle arrow is $\tau_{QD}$ (the natural transformation $\tau$ at the object $QD$) and the the outer two arrows arise by applying $F,colim$ respectively to $QD \rightarrow D$. The left arrow can be gone backwards because it is an isomorphism. The whole way from left to right is then equal to $\tau_D$ because of the naturality of $\tau$ (flip the outer arrows downwards, fill in $\tau_D$ below and you got the naturality square). This shows that each $\tau_D$ factors through $colim \circ Q$. To see that this factorization is natural in $D$, observe that $QD \rightarrow D$ and $\tau_{Q-}$ are natural in $D$.
Edit2: While Harry generously granted me a check mark after the above, Tom Goodwillie is of course right that a statement about uniqueness is in order. Here is why the above factorization is unique at the level of homotopy categories: Given any factorization $\tau=i \circ \tau'$ of our given $\tau$, for cofibrant $D$ it will factorize as
$$\tau_D=i_D \circ \tau_D^':F(D) \rightarrow colim(QD) \rightarrow colim(D)$$
where the second arrow $i_D$ is an isomorphism, because it is $colim$ of a weak equivalence between cofibrant objects. So the first factor must be $\tau_D^'=i^{-1}_D \circ \tau_D$, there is no choice here. For general $D$ we can express $\tau_D^'$ as
$$F(D) \leftarrow F(QD) \rightarrow colim(QQD) \rightarrow colim(QD)$$ by walking around the naturality square for $\tau'$ for the morphism $QD \rightarrow D$. So we also have no choice for non-cofibrant $D$.
Now I switch the notation back to mentioning the $Ho_*$: This showed that $Ho(colim \circ Q)$ is the Kan extension of $Ho_{Top} \circ colim$ along $Ho_{Top^P}$. That is already good to know, but it was just level zero of showing that $colim \circ Q:Top^P \rightarrow Top$ is the homotopy terminal homotopy invariant functor. To proceed you have to produce a simplicial set; e.g. do your localisation by passing to some model of an $(\infty, 1)$-category (e.g. hammock localisation, coherent nerve, ...), or use framings, or consider the nerve of some category of functors into $colim \circ Q$ and then show that a certain space is contractible. This should be done at a place where you can draw pictures, unlike in a MathOverflow answer.
