Since it's random now, everybody should know what to do if you are teleported.

If you are getting the relic:

Your priorities are 1. Get the relic. 2. Clear the enemies. 3. Cleanse everybody else.

If you think you will have a tough time, such as if you get teleported to the right side first (and thus have no super), throw a grenade or use a rocket to soften the minotaur up, then immediately grab the relic. If you are gunslinger, fire off three (four if you are lucky) golden guns at the minotaur, and immediately grab the relic. Then go finish them all off. For left side, I find the smash attack works well if you aim between two Hobgoblins who are close enough.

After the enemies are all dead, cleanse your teammates the instant your cooldown ends. Your cooldown starts recharging when you pick up the relic, but the Marked by the Void starts debuffing you right away. This means if you wait to get the relic, your vision can be black before you can use cleanse. Cleanse them once more when they are killing the last two oracles, then exit the portal. You aren't doing anything useful, and you want to be outside the second time's vengeance becomes active. If you are all staying in the relic shield, head immediately to the center platform or the back center. If not, hang by the portal and keep firing your super at Atheon, cleansing your teammates once they exit the portal.

If you are killing oracles:

Help soften up the enemies below if you can, especially if you are a gunslinger. Do not forget the first oracle that appears, almost right above you. There are 7 total, and they always spawn in the same order in the same places. Stay close to your partner so one cleanse can get you both. After killing the last oracle, exit the portal and get to where you can be cleansed.

The three up video I posted shows everything going smoothly.