Please ensure that you have entered delivery times in your shipping settings. This time includes the processing time and the approximate shipping time.
If zero is entered it will tell the customer that it takes zero days to deliver!
Go to: Store Manager / Shipping / Select Shipping Option and edit.
Different processing times for different products.
Create new shipping options to be assigned to products with different processing and shipping times and assign the shipping option to the relevant products.
State your processing and delivery times in your Store Policies.