It stacks
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
If you take the attack action, you attack twice.
At the start of your first turn of each combat, you walking speed increases by 10 feet, which lasts until the end of that turn. If you take the Attack action on that turn, you can make one additional weapon attack as part of that action. If that attack hits, the target takes an extra 1d8 damage of that weapon's damage type.
If you take the attack action, you can make one extra attack.
Nothing in these two statements prevents the other. One changes your number of attacks to a flat number (2), the other says "whatever it was before, +1".
2+1 = 3, you get three attacks in the first round of combat, and one of those 3 attacks deals extra damage. You get only 2 attacks for the rest of the combat.
On multiattack:
Multiattack is a monster specific action. It is NOT the 'Attack action', it's a specific action some monsters have, that determines what it can do with those attacks. This is to prevent, for example, a bandit captain from using its multiattack to make three grapple attempts in a single round. Both your features require the 'Attack action'.
The Dread Ambusher attack can be taken at any point in the Attack action after the first attack
Dread Ambusher says:
At the start of your first turn of each combat, you walking speed increases by 10 feet, which lasts until the end of that turn. If you take the Attack action on that turn, you can make one additional weapon attack as part of that action. If that attack hits, the target takes an extra 1d8 damage of that weapon's damage type. (XGE 42)
Note that nowhere does it specify any timing restrictions on the attack. Specifically, the wording does not specify or even imply that the extra attack happens after or even as a result of the Attack action. In fact, it says the attack is made as part of the Attack action.
To take the Attack action you must actually attack something, so that means you must use one of your normal attacks first. After that, it doesn't matter where you insert the Dread Ambusher attack.
This is supported by the fact that the ability says you get an "additional" attack this implies at least one other attack has already happened.
There are no rules dictating what order you must take attacks in when you have more than one in the same Action so there is no reason to think that this is not the choice of the person doing the attacking.
Extra Attack has no bearing on the ruling
In this case, Extra Attack actually has no bearing on the issue.
Extra Attack says:
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on Your Turn. (PHB 92)
Extra Attack does not insert any kind of restrictions or specifics about timing of attacks. Thus, the answer is the same if you are taking a single attack as part of an Attack action or if you have the Extra Attack feature: the Dread Ambusher attack can come at any point in the action after the first attack.
This ruling has very little impact
The only impact that I can think of for this ruling is that it allows the player to roll a potentially higher damage attack roll sooner in the turn. The only obvious advantage to which would be that it would have a higher chance of killing the creature outright and thus saving on wasted attacks.
Otherwise, there really seems to be nothing here that would impact the game in any significant way.
Best Answer
Mechanically speaking, it is not possible to attack an unaware creature outside of initiative order.
Such a situation should be resolved by an initiative roll in which the unaware creature is Surprised until the end of their first turn - and then the Dread Ambusher feature applies as normal, during the Gloom Stalker's first turn bonus attack.
Thematically speaking, if your DM adjucates surprise with an attack "outside of initiative", applying the extra d8 to a Gloom Stalker's attack is appropriate.