Here is my attempt at a partial answer:
First, a recollection about colimits along coCartesian fibrations: Let $p: X\to Y$ be a coCartesian fibration, and $f: X\to D$ a functor to a cocomplete category.
For each $y\in Y$, we can consider the colimit of the restriction of to the fiber $X_y$ of $p$ at $y$ : $\mathrm{colim}_{X_y} f$. $p$ being a coCartesian fibration means that $X_y$ is (pseudo)functorial in $y$, so these colimits are functorial and assemble into a functor $Y\to D$, whose colimit $\newcommand{\colim}{\mathrm{colim}} \colim_{y\in Y}\colim_{X_y}f$ is isomorphic to $\colim_X f$.
I'm going to apply this here to $Tw(p) : Tw(E)\to Tw(B)$, where $Tw$ is the twisted arrow category. Recall indeed that $\int^C F = \colim_{Tw(C)}F\circ q$, where $q: Tw(C)\to C\times C^{op}$ is the canonical projection.
In particular, I need $Tw(p)$ to be a coCartesian fibration. A sufficient condition for this is (if I haven't made a mistake) that $p$ be a bicartesian fibration, that is, it is a coCartesian fibration and a cartesian fibration. Equivalently, it's a cocartesian fibration and for any $b_0\to b_1$ in $B$, the induced functor $E_{b_0}\to E_{b_1}$ has a right adjoint.
In this case, we can compute $\int^E p^*F = \colim_{Tw(E)} p^*F \circ q_E= \colim_{f\in Tw(B)}\colim_{Tw(E)_f} (p^*F\circ q_E)_{\mid Tw(E)_f}$.
Let us see what this says. $p^*F : E\times E^{op}\to D$ is simply $F\circ (p\times p^{op})$ and clearly $(p\times p^{op})\circ q_E= q_B\circ Tw(p)$ so that $(p^*F\circ q_E)_{\mid Tw(E)_f}$ is a constant functor with value $F\circ q_B(f) = F(b_0,b_1)$ if $f: b_0\to b_1$.
So the colimit simplifies to $\colim_{(b_0\xrightarrow{f}b_1)\in Tw(B)} |Tw(E)_f|\times F(b_0,b_1)$ where : $|C|$, for a category $C$, is the "classifying set" of $C$, i.e. the set of path components of the undirected graph spanned by $C$; and for a set $P$ and an element $d\in D$, $P\times d$ means a $P$-indexed coproduct of $d$.
Now we're getting closer to a $B$-indexed co-end, because the $F(b_0,b_1)$ factor has the form $F\circ q_B$, but the term $|Tw(E)_f|$ doesn't seem to be so nice. We obtain:
If $p: E\to B$ is a bicartesian fibration, $\int^E p^*F \cong \colim_{f\in Tw(B)}(|Tw(E)_f|\times (F\circ q_B))$
If $p: E\to B$ is a bicartesian fibration, and if the functor $Tw(B)\to Set$, $f\mapsto |Tw(E)_f|$ is of the form $T\circ q_B$ for some functor $T: B\times B^{op}\to Set$, then for any $F$, $\int^E p^*F \cong \int^B T\times F$.
Note that this assumption on $Tw(E)_f$ is "universal" and doesn't depend on the functor $F$. This is somehow because there is a universal coend.
Indeed, if you think about the case where $F: B\times B^{op}\to Set$ is constant with value $*$, then you'll find that $\int^E p^*F$ is going to be exactly the colimit over $Tw(B)$ of $Tw(E)_f$, so if you want a general co-end formula, it better specialize to one in this case.
So when is this hypothesis satisfied ? I'm not sure. Certainly, $f\mapsto Tw(E)_f$ actually depends on $f$ and not just $b_0\to b_1$. In terms of objects, letting $f_!$ denote the pushforward along $f$, an object of $Tw(E)_f$ is classified by some $e_0\in E_{b_0}$, some $e_1\in E_{b_1}$ and some $f_!e_0\to e_1$ in $E_{b_1}$.
Naively, it looks like $Tw(E)_f \simeq E_{b_0}\times_{E_{b_1}}Tw(E_{b_1})$ which looks like it doesn't depend on $f$, but $E_{b_0}\to E_{b_1}$ in this pullback is precisely $f_!$. Maybe in certain situations you can prove that it doesn't depend on $f$, but this would be specific to certain $p$'s. For example if $B$ is a preorder then there is at most one $f$ per $b_0, b_1$, so that $Tw(B) \to B\times B^{op}$ is a fully faithful inclusion, so there exists a factorization through $q_B$ (say by left Kan extension). Therefore, a special case is :
If $E\to B$ is a bicartesian fibration, and $B$ is a preorder, then $\int^E p^*F \cong \int^B T\times F$, where, if $b_0\leq b_1$, $T(b_0,b_1)\cong \colim_{(x,y)\in E_{b_0}^{op}\times E_{b_1}}\hom_E(x,y)$.
Let me also finish with a remark: I said bicartesian fibration to have that $Tw(p)$ was a coCartesian fibration and have this nice formula with $Tw(E)_f$, but the recollection I gave at the very start works for an arbitrary functor $p: X\to Y$ if one replaces $X_y$ with $X_{/y} := X\times_Y Y_{/y}$. In our story we then get the same thing with $\colim_{f\in Tw(B)}\colim_{Tw(E)_{/f}} F\circ q_B\circ Tw(p)$ except that now the latter isn't even constant on $Tw(E)_{/f}$. If you assume on top of everything that it is, then you can again do something similar etc. But this is going to be even rarer.