It takes some time for the sale to start and stop because Product Discount sends a series of requests (API calls) to Shopify to get the product, set the compare at price, change the product price, and a few other tasks. Shopify has an API call limit, throttling how many updates can process at a time.
In general, it takes roughly 1 hour per 1000 variants to apply the discounts.
If you have 1000 products with one variant each or 100 products with 100 variants each, it will take about an hour to start the sale, and another hour to stop the sale.
Can you make it go faster?
Shopify's platform puts restrictions on the number of requests that a store can make in a given period. They also have request queueing on their side, so even if we push the data faster, it could take an extended period for their servers to process the request.