No

But the argument is not so straightforward.

The problem with this question is that the RAW are murky enough to allow for some leeway in interpretation. Going by RAI, the intention was clearly to use Two-Weapon Fighting with one-handed weapons, and not game the system by attacking with a versatile weapon while holding it in two hands, arguing that it still counts as a one-handed weapon for the Dual Wielder feat.

The Rules

Dual Wielder

You can use two-weapon fighting even when the one-handed melee weapons you are wielding aren’t light.

While this qualifies the longsword for two-weapon fighting, the Versatile property states that:

Versatile

This weapon can be used with one or two hands. A damage value in parentheses appears with the property—the damage when the weapon is used with two hands to make a melee attack.

This explicitly states the need to use both hands to make a two-handed attack and use the bigger weapon dice, thus disqualifying this use case from the Dual Wielder requirement (one-handed melee weapons).

The Conclusion

While you can use a thrown handaxe and then attack with the longsword, you can only do so using the one-handed dice the Versatile weapon tag provides.

Ability Score Improvements are almost always better for DPR

Calculation

On 4th level you could raise your attack stat from 16 to 18, or take Dual Wielder. I will assume you have the Two-Weapon Fighting style, while fighting against AC 10, 14 and 18. Criticals are also included in the final DPR.

Base DPR:
Damage 2 x (1d6+3) = 13
AC10: Hit chance 80%, 10.75 DPR
AC14: Hit chance 60%, 8.15 DPR
AC18: Hit chance 40%, 5.55 DPR

ASI:
Damage 2 x (1d6+4) = 15
AC10: Hit chance 85%, 13.10 DPR
AC14: Hit chance 65%, 10.10 DPR
AC18: Hit chance 45%, 7.10 DPR

Dual Wielder:
Damage 2 x (1d8+3) = 15
AC10: Hit chance 80%, 12.45 DPR
AC14: Hit chance 60%, 9.45 DPR
AC18: Hit chance 40%, 6.45 DPR

You can look at it this way, the damage is the same, as on average 1d6+4 = 1d8+3, but after the ASI you have higher hit chance.

Other considerations

Dual Wielder gives you +1 AC too, but if you are Dex primary, an ASI gives you the same, and +1 save, +1 skills, +1 initiative.
If you are Str primary the ASI is much less helpful in this regard. (But in this case you could just take Great Weapon Fighting, for significantly higher DPR)

Dual Wielder lets you draw two weapons without sacrificing an action. This can be important in cities, but in the wilderness and dungeons you can just run around with one of your weapons already drawn.

Exception

Take Dual Wielder if the only magic weapon you have is unusable without Dual Wielder (because it's not light).
+1 to attack and damage from a magic weapon plus the increased die size is better in itself, but if you consider all the monsters resistant or immune to mundane weapons, it is the highest priority.

Conclusion

You should only pick Dual Wielder if:

• the only magic weapon you have is unusable without Dual Wielder
• you can't keep at least one of your weapons in your hand when you expect a fight
• you cannot pick an ASI, because
  • you are a Variant Human on 1st level
  • you have reached 20 in your attack stat already