I'm willing to work in the category of compactly generated Hausdorff spaces.
The ordinary geometric realization of a simplicial space commutes with taking pullbacks and preserves the terminal object, hence commute with arbitrary finite limits. (A reference with a proof is Corollary 11.6 in May's 'Geometry of Iterated Loop Space')
It is stated in various places (Proposition 9 on this page of the nLab, Remark 2.23 of 'Homotopy Theory of Orbispaces' by Gepner and Henriques) that the fat geometric realization preserves finite limits up to homotopy, but I could not find a solid reference for a proof of this statement. Since it is obvious that it preserves the terminal object up to homotopy, it suffices to have a proof for the pullback-case.
If all simplicial spaces in consideration are 'good' then the statement follows from the preservation for the ordinary realization, but I'm interested in the general case.