First of all, forget about getting this on Hard. You will have almost no advantage over Vek already because of the time limit, so competing against larger and more powerful force while also trying to get this Achievement is a guaranteed suicide.
Put Camila Vera inside your Lightning Mech, possibly with extra Core (spend it on the 'Building Chain' upgrade immediately). Being immune to Webbing and Smoke helps a lot, as it lets you bait the Vek to attack your mech instead of Civilian Buildings.
Do not try to get this achievement if this is your first time playing the Blitzkrieg Squad. Familiarise yourself with their abilities and positioning specifics before attempting to get the achievement.
Choose Islands with Spider or Goo Bosses, as they syngergise with Lightning Mech's abilities best and allow you to keep emerging enemies at bay.
For practice purposes, try to complete the first 2 Islands without getting any new weapons. You will be lucky to have any Reputation left for spending on weapons after completing the 1st island, as your main priority for getting this achievement is to keep your Grid Power high enough.
Prioritise protecting Civilian Buildings over anything else. The mission tells you to defend the train? Forget about that, it's a free building decoy now.
Don't chase high-Reputation missions on 2nd Island. You won't have a chance to spend it anyway.
Don't spend too much time in the Corporate Store. Restore the Grid Power to an acceptable level and proceed to the next Island. The time you spend on deciding whether or not to buy that new weapon is the time you won't have to complete the challenge.
Turn on the timer in settings. This will allow you to track the time left.
Turn up enemy movement speed in settings. This will allow to save a lot of time.
Take 'Defend the train' missions where available. They last one turn less than other missions.
You only need to complete the first two Islands (any) for this achievement to trigger. Winning the game is not required.
As soon as you get into the victory screen for the 2nd Island, the achievement will trigger, but not before. Take this into account, as Island victory screen appears with ~5 second delay after completing the HQ mission.
You have 10 missions (4 generic missions + defending the HQ on each island) to complete in under 30 minutes, which roughly estimates to 3 minutes per mission.
The timer stops when you enter the mission menu (pressing Esc).
If you use a reset, the timer will be rolled back to the beginning of the turn.
You can take a screenshot of the battlefield and hit Esc to evaluate your next moves. Note: this is considered a dirty tactic that is usually frowned upon, as it completely defies the purpose of the achievement.
If everything else fails, turn down the Difficulty and get the achievement on Easy.