Was the 30-06 Springfield used on the British side in World War 2?

No. The standard British infantry rifle was the Lee-Enfield in .303 caliber in both world wars. During WWI several million of these were produced for Britain by US civilian firearms manufacturers, in .303. When the US got into WWI they still had the dies and machinery, and produced millions for the US, after slight adjustments, in the standard us caliber .30 (".30-06") so they took the same ammunition as the 1903 Springfield. More WWI troops had one of these "American Enfields" than had a Springfield. The British never used the 1903 Springfield, other than perhaps a few here and there on an individual basis.