Both these sources say the nature is determined when you interact with the TV.
Some places say it can be changed if the lead Poke has Synchronize, but...
Also from Emerald onwards, if the Pokémon with the Ability Synchronize is leading the party, there is a 50% chance of encountering a wild Pokémon with the same nature. Synchronize does not have this effect on stationary legendaries on Emerald...
So, if you chose Red, It is stationairy and cannot be changed. still working on when it is determined.