[handwave] The 1948 war goes badly very quickly for the nascent Israel, the Jewish militias are subdued quickly because some crucial supplies which were there OTL, well, weren't there. [/handwave]
The bloodshed stays within limits. The Arab forces draw plans to deport or disperse the Jews, but then, before any of this gets implemented, things sour between Syria and Lebanon, on the one side, and Egypt and Jordan on the other side. In the face of possible war, Egypt decides to offer citizenship to Jews under condition that Jewish veterans fight for Egypt, against Syria.
Maybe it comes to blows, maybe they just stare at each other, in any case the Jewish officer corps becomes an integral part of Egyptian army - especially if the Egypt/Syria divide is Cold War motivated, this may get Egypt additional support from France and US.
This is the environment in which Netanyahu makes his military career. After distinguished service in the one or another war Egypt participates(hmmm.... *throw a dart into a map* Sudan) , he climbs the military career ladder almost to the end - until the Soviet Union collapses in a similar manner as OTL. Soviet support for Syria dries up, and as the previous Cold War frontier becomes a political backwater the same happens to the Western support for Egypt. The economical situation worsens to a stage where riots occur, radical parties of all shapes (from Islamist to Communist) fight for the control of the streets, and so on. In this climate, a small but determined group of generals overthrow the paralyzed government, and imposes a "temporary" military rule to "restore order". Head of the said group is the distinguished Sudan War hero, the General of Artillery Binyamin al-Netanyahu...