Why would you expect them to flash after uploading is finished? Obviously they do during the upload, because that is serial activity.
When the upload finishes, your sketch starts. There will be serial activity only if your sketch generates it - and if there is some process on the PC to which it communicates, such as the "serial monitor".
Similarly, Pin 13, the "L" LED flashes only if your sketch tells it to do so - such as the "blink" exercise. Not for any other reason though on a UNO R3, the "L" light may randomly be on or off if your sketch has not initialised that pin as an output and it is not otherwise connected.