According to RAW, it seems to be possible to interpret that one doesn't need the Dual Wielder Feat to attack 3 times

Quote from Kchaka >>

Since the last sentence that says "only once per turn" is written in the Nick Mastery and not the Light Weapon Property, I think it's reasonable to interpret it as "You can make this specific "converted" extra attack only once per turn.", referring to benefiting from this Nick Mastery conversion only once per turn.

If the intention was to create a restriction on how many times per turn one could "benefit from this Nick Mastery conversion", then the text would have said exactly that.  It doesn't.  Reading it that way is invalid -- the text simply does not say that.  Instead, it creates a restriction on how many times per turn you can make this extra attack.

Note that there might still be additional triggers that enable this extra attack.  It's just that you cannot actually make this extra attack if you have done so once already on this turn.  This restriction applies immediately upon executing the text for the Nick Mastery property feature.  You don't get to pick and choose which parts of a feature apply.  When you use the Nick Mastery property that restriction gets put into place and applies for the rest of that turn.

It's basically a resource issue.  It's sort of similar to a situation where something happens in the game that triggers the possibility of you using your Reaction to cast the Shield spell.  However, if you've already used your Reaction then you are unable to actually cast this spell at this time.  This limitation doesn't have to be written into the spell description of the Shield spell.  This rule exists elsewhere, but it applies anyway.

Quote from Kchaka >>

Also, the wording of the Enhanced Dual Wielding from the Dual Wielder Feat is almost exactly the same as the wording from the Light Weapon Property

Yes, this is intentional.  It provides a separate extra attack which is somewhat similar to the one that is provided by the Light property.  "This Extra Attack" does not refer to the extra attack of the Dual Wielder Feat.  It refers to the extra attack of the Light property.

Although these mechanics are arguably confusing, they are not actually ambiguous.  There is exactly one correct interpretation.