If a player discovers both alien species in a single turn (for example, by completing the third discovery space for both via a milestone), the order in which the species are revealed and rewards are given is as follows:

• Milestones are resolved between turns, after the current player has finished playing free actions (page 18).
• If multiple players have reached milestones, they resolve them in order, starting with the player whose turn has just ended, and proceeding clockwise. The neutral milestone belongs to nonplayers, and it is always resolved last (page 18).
• If a species is discovered during your turn, resolve the discovery after you finish your turn. (If there are any milestones, resolve them before resolving the discovery.) (page 20).

Summary

• First, resolve any milestones triggered (including neutral milestones that may fill discovery spaces).
• If both species are discovered as a result of these milestones, resolve the discoveries after all milestones are resolved.
• The rulebook does not specify a required order for revealing the two species if both are discovered simultaneously. However, since both are discovered "after you finish your turn" and after milestones, you may choose the order in which to reveal and resolve the discoveries.
• For each species:
  • Flip the species board over to reveal which species has been discovered.
  • Read the rule sheet for that species.
  • Follow the setup rules for the revealed alien species.
  • The players that marked the three discovery spaces are rewarded at this time.

Source: page 18 (Milestones), page 20 (Discovery process and timing). The rulebook does not specify a strict order if both are discovered at once, so either order is permissible as long as both are resolved after milestones.

Source: Core Rulebook (pp. 18, 20, 26, 28, 3)