not sure how the timing works here.

Yes, it has nothing to do with timing. If you play the card while R2 is alive then you gain control of the terminals until the end of the round. This effect doesn't go away if R2 is destroyed and he is not required to be on the board anymore once you have played the card.

The card reads:

Use while adjacent to a terminal. Until the start of the next round, you gain control of all terminals, regardless of which figures are adjacent to them.

The confusion is about "You". But this can be solved by looking at the RRG under "Control":

Many missions require players to control tiles or tokens.

Only players can control tiles or tokens. So the "you" on Terminal Network does not reference R2-D2 and thus the effect continues, even if he is defeated.