Actually, I would suggest this would have profound impacts. Most scholarship suggests that the destruction of the temple had profound impacts on the emergence of "Rabbinic Judaism". Specifically, the Temple formed a key aspect of Jewish religious practice and particularly the way to atone for sin. When the Temple was destroyed this created an "existential" question for Judaism. Indeed, there was a plausible argument that without the Temple Judaism could not exist. Over the succeeding generations attention was focused on developing the Talmud and focusing on a faith of the mind rather than a faith based on animal sacrifice. I would suggest this occurred because there was no alternative. If the temple is rebuilt in say 120 CE you know have serious competition. More traditional Jews move away from Rabbis and toward the Temple (worshiping as their grandparents had done). Arguably the Talmud is not completed or is completed in a very different form. It may also strengthen Jewish-Christians (i.e. the followers of James the Just of Jerusalem) relative to the Gentile Christians of Peter and Paul. If the followers of James can return to the type of Temple worship that the Gospels discuss Jesus engaging in this may keep them from dying out.
Now, on the question of how plausible it is for Hadrian to actually rebuild the temple, I think that is a different question. However, given the assumption that he does, the impacts will be profound.