The Assault Jets and Blink Pack are very useful for this achievement, as they can be used to jump over the biomass and skip the bridge as soon as it is activated.
You don't have to destroy the biomass blocking the bomb in order to arm the bomb.
A Chainsaw wielded by Vegas is very useful, as it breaks rocks and doors 8.8% faster than the Mining Laser does (excluding the time required to equip and charge the laser).
The Grenade Launcher is the best weapon for quickly destroying the Eggs near the bomb-arming terminal.
The Hornet Barrage or Smart Bomb can help you handle large groups of enemies. One of the best places to use them is for dealing with the multitude of eggs (and any Parasites that hatch) at the computer terminal near the end.
A total of 6 Parasites will spawn on the moving bridge after it rotates, and 3 more will spawn as you run back. Note that the Parasites on the bridge first spawn when the squad starts crossing the bridge; if a marine uses the Assault Jets or Blink Pack to enter the bridge as the rest of the team starts crossing, the Parasites may spawn on top of the marine.
All living marines must reach the landing pad before the ship begins its descent, so rushing ahead of teammates to the extraction point will not speed up the run.
Before you reach the Blink Pack, activate your Smart Bomb to assist your teammates. As soon as your team breaks through the pile of rocks, teleport on top of the Parasite at the top-left corner of your screen and destroy the door ahead using both weapons (disregard the stunned Parasite). You are now the only player who will determine your team's time for the rest of the level.
Pick up the Assault Jets and reach the northern pile of rocks as demonstrated (this video demonstrates how to make the first two jumps). Saw through the rocks, ignite all of the Eggs in the area, and kill the incoming Drones. Have your team activate Adrenaline, and quickly hack the computer; strafe during the hack if necessary to prevent the Drones from interrupting it. Rush back and pick up the Blink Pack (transported by a teammate) where the biomass used to be. While hurrying back to your teammates, teleport onto the rotating bridge, and also teleport onto the top of the stairs at the landing pad to initiate the landing sequence asap, as you should be the last person on the landing pad.
For a slower but easier alternative to hacking the terminal with incoming Drones: Upon reaching the northern pile of rocks, break the foremost rock and, without moving forward, jump in front of the mast next to the staircase as seen here. By doing this, no Drones will spawn near the terminal and you can safely hack after destroying the eggs.
Use grenades to dispatch the Harvester Spawn and Drones in your way up to pile of rocks. Destroy the rocks with the Chainsaw and all of your Hornets, and kill the stunned Parasite ahead. Help Vegas destroy the door ahead, activate the rotating bridge asap, and pick up the Blink Pack. If you can manage the jump, use it to get a head start. Otherwise, meet up with Faith and Jaeger. When you reach the biomass, juggle the Blink Pack into Vegas' path by exchanging it with the nearby Healing Kit.
When you reach the pile of rocks, help Wildcat get through.
Stay as close as possible to Wildcat and Faith so that you can cover them while they destroy the rocks. As you wait for the rotating bridge, kill the enemies on the west side. When you get near the biomass, set down (but don't build) your Sentry as shown, sustain fire on the left section of the biomass, and lay trip mines in the pattern shown in Vegas' perspective as he backtracks.