The discount rule activation time depends on quantity of products. Only one discount rule can be in process at a time while the rest will be under pending status.
The time taken for discount rule activation (or deactivation) is about 2 products per second, i.e. it takes 5 seconds to apply a discount to 10 products.
To know if the discount rules had been activated, the "Active from <time> <date>" status will be displayed at the discount rule: