When someone sent you a letter, you will receive a notification immediately.
However, just like a real letter, delivery time in Slowly depends on geo-distance between pen friends. You are not able to read the letter until it arrives your location.
You can check the remaining delivery time of the incoming letters in Slowly map, by tapping the inbox button in Home screen (under Recently Received).
Once your letter has arrived, you will receive another notification. Then, you will be able to read the letter which can be found in Recently Received, or inside your Pen Friends list.