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.