At the end of the first level, you will need to make a decision that slightly affects how the rest of the game plays out. One of the options will lead to saving Fergus, and thus this achievement; the other will net you
I would recommend going for this achievement (Fergus) in your first playthrough.
The differences between the timelines are lock-picking and armour upgrades if you save Wyatt, and hot-wiring and health upgrades if you choose Fergus. There are also a couple of different people to talk to in some of the bits between major levels.
The reason I suggest saving Fergus in your first playthrough is the health upgrades. On many levels you can get a permanent +10 bonus to your health; the equivalent on the Wyatt timeline just increases the amount of armour you get from each armour pickup, but the 100 total can't be increased, and armour is plentiful on most levels. There are 10 health upgrades over the various levels, which can thus give you a base health of 200.
This can carry over into your Wyatt timeline playthrough if you use the Chapters option instead of starting a New Game -
selecting New Game will reset your progress on everything, including collectibles and perks, but you can use the Chapters option to jump into the alternative timeline with all perk and collectible progress retained.
EDIT: It would appear that the 'Wyatt Saved' achievement is glitching for some people. The achievement tracking on Xbox One is reading 1 of 1 requirement completed, but it is remaining locked for unknown reasons. I believe (and hope) that this will rectify itself in time, as the systems is clearly showing that the gamer has done what is required, and there have been numerous cases of X1 achievements unlocking days or weeks after the requirements have been met.
All the same, be aware that this has happened to some gamers, and consider saving Wyatt first, then immediately replaying the first level to save Fergus and then continuing that timeline for the benefits outlined above.
EDIT #2: This has now popped for everyone who I'm aware of it glitching for, so hopefully the issue has been fixed going forward.