Here is the full list of all 14 Nebula Nuker achievements.
Destroy the Obliter
Destroy the Purity Nexus
Destroy the enemy homeworld
Collect a powerup
Purchase an upgrade
Clear a level using only turrets
Destroy 4 ships with 1 diamond-cannon powerup
Destroy 100 enemy ships
Destroy 1000 enemy ships
Purchase all upgrades
Purchase a consumable
Fill up consumable storage
Purchase or unlock a new skin for your ship
Have access to five different skins for your ship