It was not a religious conflict. It was primarily a conflict between competing nationalisms that was coopted by other surrounding states and larger powers as it began.

Zionism began as a movement for Jewish self-determination and statehood. It has roots in the longstanding Jewish religious yearning for the arrival of the messiah, in some interpretations, with a whole bunch of other facets that are complex and beyond the scope (from what will happen to when and in what order).

Broadly speaking, however, Jews have continuously prayed towards Jerusalem. They maintain the hope of return to Jerusalem in their religious services, and have for millennia. In the context of rising antisemitism, after centuries of attempting to both separate from antisemites by forming insular communities and also attempting to assimilate when those insular communities were attacked, Jews were swept up in the tide of "self-determination" that began to sweep the world post-Enlightenment in particular. This eventually culminated in the Zionist movement, which sought to bring Jews to join the Jews already in the area now called Israel, which then was part of the Ottoman Empire (and before that was held by Arab empires, and before that other empires, etc.), and form a state there.

In conflict and tension with this were other forms of nationalism. As the Ottoman Empire weakened, and eventually fell apart following WWI (leaving the area in British hands, who called it the Mandate of Palestine and agreed Jewish self-determination should be exercised there alongside Arab self-determination), other nationalisms especially continued to grow and take root from smaller forms as well. Some of these were pan-Islamic, and others were pan-Arab, while others were more regional (pan-Syrian, for example), and some were more local. During and also following the collapse of many of these ideologies as other Mandates under French and British control post-WWI came up in the area, and eventually became states like Jordan or Syria, the only ideology that remained standing was a Palestinian Arab one, local to that specific British Mandate for Palestine. That identity and ideology of national self-determination was itself distinct, and overlapping with others. If that sounds confusing, that's because it is, but it shouldn't necessarily be. One can be both a Wisconsinite, a Midwesterner, an American, and a Milwaukeean at once, while also being a Protestant Black individual. These identities all overlap, and in a situation where none is predominant (or there is conflict among them and with other outside ones, i.e. Zionism), there is bound to be a push towards general resolution, at least in my view, which helps frame what happened.

So when Israel declared independence, it was already in the midst of conflict. Not just of ideologies, either; for decades already, there had been fighting between these ideological outcroppings. There were debates among communities, and some violence, notably for example riots in 1920, 1921, 1929, and an Arab revolt (in part against the British rule, and in part against Jewish immigration/self-determination desires, among other factors) in 1936. From 1945-47, there was also additional continued fighting between Zionist and Palestinian Arab militias and groups, though both also fought the British military (more the Zionists by then, I'd argue, as Palestinian Arab militias were still recovering from the damage inflicted on them during the Arab Revolt). But since the British were essentially unwilling to continue trying to find a solution that pleased both Arab and Jewish groups, having proposed partition and one-state solutions to no avail, the United Nations investigated potential solutions via a community composed of non-superpower representatives. Those delegates visited the region, met with Jewish representatives and Arab states around the region, as well as some Palestinian Arabs (despite the official Palestinian Arab position of boycotting the committee, viewing it as unfair because they did not recognize any legitimacy to the Jewish claims). They eventually recommended a partition, which was recommended in a nonbinding UN General Assembly Resolution in 1947.

This is when the war actually began, contrary to common misconception. Following the approval of the resolution, Arab militias appear to have struck first, targeting some Jewish buses, and the resulting spiral of escalation led to an outright civil war. The Jewish forces were outnumbered but better trained, despite Arab states sending "volunteers" and weapons to help Arab militias. They had all but won when Israel declared independence in May 1948, on the day the British Mandate formally expired (thus ending British "rule", though troops would continue evacuating through August). It was then that the Arab states invaded.

So the conflict was much more about conflicting nationalisms than religion. Religious conflict can and does play a part in the later stages of the conflict, but is far more complex, and doesn't really explain the start. Similarly, superpower dynamics and the Cold War played their own part, but that largely came post-conflict-outbreak, much moreso than prior.