Yes. A sketch:
Taking products with the free $G$-space $EG$ commutes with the pullback diagram (because product is also a limit) and so you can assume they're free, and one of the maps is a fibration.
Having done this, there is a natural long exact sequence of homotopy groups
$\to \pi_* (U) \to \pi_*(U_{hG}) \to \pi_*(BG) \to \dots$
and applying this to the pullback diagram you can deduce (from the 5-lemma) that the natural map from the orbit of pullbacks to the pullback of the orbits is a weak equivalence.
