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Philippe Gaucher is right. This problem was solved by Julie Bergner, here. I recently asked a question that summarized some of her work on this problem. The point is that the homotopy limit of your diagram is a category $M$ whose objects are 5-tuples $(x_1,x_2,x_3,u,v)$ with $x_1 \in C'$, $x_2 \in D$, $x_3\in C$, and $F(x_1) \stackrel{u}{\to} x_3 \stackrel{v}{\gets} G(x_2)$ in $C$, where $F$ and $G$ are the two functors in your diagram. The morphisms in this category of 5-tuples are obvious. This category $M$ can be given a model structure where the weak equivalences and cofibrations are levelwise (on each $x_i$), and that model structure can be localized if desired to force $u$ and $v$ to be weak equivalences in the local objects of $M$. Bergner then proves $M$ has the correct homotopy type, meaning that, upon passage to complete Segal spaces (i.e. $(\infty,1)$-categories), it becomes the actual homotopy pullback of the diagram. She has to assume the model categories she starts with are combinatorial, but this seems a standard assumption now from the $\infty$-categorical perspective (i.e. assuming presentability). Bergner uses a right Bousfield localization, so you need to assume right properness, or pass to right semi-model categories like Barwick does in this paper. The difference between a semi-model structure and a full model structure is invisible to the underlying $(\infty,1)$-category.