The code you linked to is an Atmel Studio V7 project,
so there will be some work to be done, before it works in the Arduino environment.
I've not bothered with the serial upload to a Nextion, I always used the micro SD.
So I would prefer to extend the SD to the outside of the enclosure,
or have it accessible in the first place.
You could use a multiplexer for the serial pins, you don't have to use a relay.
I hope you took into account, that the Nextion is a 5V device and the DUE only supports 3.3V.
So a voltage divider on the Nextion's tx pin is needed for in-spec operation.
Just out of curiosity, why do you expect the GUI to change after deployment?