How could a Jewish State fight a World War II?
Let's assume there has been a much larger influx of Jews into Ottoman Palestine in the 19th Century, which has been allowed by the Ottoman authorities. After the dissolution of the Ottoman Empire, with the encouragement from the British, the Jewish State declared its independence concurrently with the Arab States, was recognized by the later as an independent state with West Jerusalem as its capital.
All ME states, Jewish and Arab, were still under British imperialism as in OTL, but the status of a Jewish state in the region has been undeniable by the 1920s in this universe. By late 1930s it had almost gained full Self Rule, similar to Canada and South Africa, with British troops still stationed in some isolated bases.
The Nazis still took power in 1933, and the World War still took place in 1939.
Now, how would this Israel react to the Antisemitic Third Reich in Europe?
It could send its own expedition forces to Europe, alongside the BEF;
It could assist the Balkan countries against the Nazis;
It would become a prime target of Rommel's Afrikakorp, and had to defend itself;
It could also send a Expeditionary Force to the Soviet Union, just like the Normandie Niemen Regiment.
What kind of weapons they would possess? British of course, but a autonomous state with its own military could make and import its own weapons.