In the case of Christianity Romans saw the potential of how a people can rally behind one deity as did the Jews against Rome and put up considerable resistance as a result; making it eventually politically prudent to use it to unite a very diverse empire.
Islam is just a customized form of Nestorianism and Muhammad used it to similar ends. The Arabs were scattered tribes, and he pulled them all under one banner for political ends.
Judaism does have political ends, and was from time to time taken up and modified for various political ends but within Israel.
Judaism could never spread because it's not possible to convert fully. You can join the religion certainly, but you can never be a real, ethnic Jew and unless it was a gentile man with a Jewish woman your children couldn't be either (the traditional forms of Judaism state that you must have a Jewish mother to be Jewish).
Islam and Christianity just took the biggest entry restriction off. Even before Christianity, in Rome Judaism was viewed with a reasonable degree of respect to the extent there are records of Pagan Romans employing Jewish exorcists and sometimes teachers to instruct them to become "God Fearers" (that is, non Jewish devotees of Yahweh).