IMPORTANT: There are people who have reported that this achievement doesn't pop as mine did at the credits. My advice would be to do this when connected to Xbox Live, and if it doesn't pop at the credits, play through the campaign again and wait 72 hours for XL to sync. Make sure you do this in one save file, go on Hardcore, and don't kill anyone with a Lancer or Gnasher, including melee/ Chainsaw. Further clarification is given below.
As of the January update for Gears 4, you now will see chapter progress on the achievement. Each chapter you complete should go some of the way to getting the achievement. You can use the achievement tracker to make sure it is calibrated. There has been a person who says that the progress is slow to update, and thus 2 chapters progress may be put on instead of 1. Make sure you account for that if you decide to use this method.
Just to add clarity to the description, you can hold a Lancer/Gnasher, and shoot from a Lancer/Gnasher, but you CANNOT KILL AN ENEMY OF ANY DESCRIPTION WITH THE LANCER/GNASHER.
You can kill with any other gun, the Retro Lancer, Marksa, Overkill, Boomshot, Torque Bow, Grenades, Pistols etc.etc. just make sure you don't kill an enemy with the Gnasher/Lancer, play on Hardcore difficulty (or harder if you feel like challenging yourself) and the achievement should pop at the credits.
I would advise people to drop the Lancer and swap it for a Hammerburst or a Retro Lancer, if available, and swap out the Gnasher for any power weapons, like the Longshot, Boomshot or Dropshot.
In the first part of the prologue, swap out the Gnasher for the Marksa Mk.1 after defeating the first UIR soldiers you meet. You will find ammo for this from fallen enemies and from ammo boxes.
In the second part of the prologue, swap your Gnasher for a Hammerburst after you defeat the Locust Drones, again, you can find ammo from the boxes at the House of Sovereigns and from fallen Drones. Boomshots are available from fallen Boomers that you kill.
In the third part of the prologue, defeat the 2 Locust Drones at the top with your Snub or Frag Grenades. Then, pick up the Hammerburst, and there should also be a Boomshot in that area as well, which you can swap for the Lancer and Gnasher.
At the beginning of Act 1 Chapter 1, if you are playing as JD, you will start with a Marksa and a Gnasher, you can swap this for a Retro from Oscar, a Longshot from Kait, or a Hammerburst from Del.
In Act 2 Chapter 2, All Geared Up, you will automatically pick up a Custom Lancer and a Gnasher. Drop these before leaving the area, as leaving the area will trigger DeeBees to come down from the dropship. There is 2 Longshots, for 16 bullets, Hammerbursts, Marksas, Retro Lancers, Frag and Incendiary grenades to pick up. After this you won't be reset of your weapons until the very end, in Act 5 Chapter 3 when you go into the Giant Mechs, and by that time you cannot hold a Lancer or Gnasher for the rest of the game.
Be careful not to pick up the various amounts of Lancer and Gnasher leftovers on the way.
EDIT: I was presuming that the majority of people had already beaten the campaign before attempting this, or at least I would recommend you beat the campaign on Hardcore before attempting this.
It really isn't that difficult. If you are new, or struggling, look on the solution for the Insane campaign, for a more definitive guide on beating the campaign. They contain videos and tips on certain parts of the campaign. I wouldn't recommend beating the campaign without a Lancer or a Gnasher on a first time playing really. Play the campaign, learn where to go and where the weapons are, and swap them out. It isn't as difficult as Inconceivable or Insane by a longshot.
For those doing Co-op, it can be obtainable doing it in this mode, however, your partner must also not kill anyone with a Lancer or the Gnasher, and playing on Hardcore or they wont get the achievement themselves. EDIT 2: You can go on Hardcore and your Co-Op partner on Casual, and as long as the person playing Hardcore doesn't kill an enemy with a Lancer or a Gnasher, that person will get the achievement, and not the person getting Casual.