Many things could happen.
For starters, the Egyptian army wasn't that good. Poorly equipped and trained (compared to the Germans and the post 1940 Italians), they would have a hard time against the Axis and might allow the Axis some easy penetrations. If employed in rear areas, they could release British/Commonwealth troops for frontline duties however, strengthing the British position.
At the political level, assistance by the Egyptian army would have to lead to political concessions by Egypt's occupiers. The British would have to allow the Egyptians greater say in strategy as well since it is "their" country.
And if the young Turks in the Egyptian army, who OTL were plotting the overthrow of the British sponsored stooge and their own liberation, still decide to continue their freedom activities, the British position may well be undermined at important moments by Egyptian forces.