Analyzing the information about the mastery rule and talent:
1- Mastery: Nick. When you make the extra attack of the Light property, you can make it as part of the Attack action instead of as a Bonus Action. You can make this extra attack only once per turn.
2- Dual Wielder
General Feat (Prerequisites: Level 4+, Strength or Dexterity 13+)
You gain the following benefits.
Ability Score Increase. Increase your Strength or Dexterity by 1, to a maximum of 20.
Enhanced Dual Wielding. When you take the Attack action on your turn and attack with a weapon that has the Light property, you can make one extra attack as a Bonus Action later on the same turn with a different weapon, which must be a Melee weapon that lacks the Two-Handed property. You don't add your ability modifier to the extra attack's damage unless that modifier is negative.
Quick Draw. You can draw or stow two weapons that lack the Two-Handed property when you would normally be able to draw or stow only one.

Using two light weapons with the mastery described, would I have an extra attack from the mastery and an extra attack from the talent? Would Lvl 4 have 3 attacks?

I understand that the light property grants me one attack and the talent grants me another.

Yes, as a level 4 Ranger with the Dual Wielder feat, you would be able to make two attacks with your action, then make another attack using your bonus action thanks to the Dual Wielder feat. At level 5 when you get Extra Attack, you would be able to make 4 attacks total (3 with your action, 1 with your bonus action).

Also of note, if you also took the Two-Weapon fighting style feat, you would be able to add your modifier to all of these attacks.

Yes, but 
* the light weapons rulelets you make two attacks if you are wieldng 2 light weapons but the second attack normally uses your bonus action.
* the Nick mastery property allows you to move the Nick weapon attack out of the bonus action into the regular attack action. This frees the bonus action for things like spell casting ( the ranger’s hunters mark for example) or other bonus action activities.
* the dual wielding feat allows you to make a bonus action attack with a light/one handed/one handed versatile weapon if you are using a light weapon for the primary attack action attacks.
This means that at L-3 if you wield 2 light weapons with be being a Nick weapon you get 2 attacks in your attack action and still have your bonus action available. At L4 if you take dual wielding you can add a third light weapon attack as your bonus action or ( depending on the weapon switching shenanigans your DM allows) a bonus action attack with a non light weapon after your single light weapon attack action attack or 3rd attack with you non light weapon after 2 light weapon attacks in your attack action. From L5 on if  you get extra attacks and mastery properties you can potentially get 4 light weapon attacks or 2 ( or 3) light weapon attacks and a non light weapon attack. 
By RAW you get one sheathing/drawng action automatically then dual wielding grants either 1 or 2 more depending on how your DM reads iit and dropping a weapon is now a sheathing not a free action. So switching back and forth between light and non light weapons rapidly becomes unsustainable by RAW.  So at L5+ your really better off with just 2 light weapons getting 4 attacks ( avg damage =14) than using a D8 weapon ( avg damage 3 attacks = 11.5) unless your DM allows you to play switching games without restriction (avg damage =15)

I don't know what "shenanigans" you're talking about with regards to sheathing/unsheathing and getting extra sheathing/unsheathing actions as part of Dual Wield? As part of the Attack action, you get to equip or unequip one weapon, either before or after the attack. This does not mean that you get to equip or unequip a weapon every time you attack. The Dual Wielder feat allows you to "draw or stow" two weapons when you would normally be allowed to draw or stow one, NOT to equip or unequip additional times. So you can't attack with one weapon, then stow it and draw another unless you have an ability that allows you to specifically make two Attack actions, like a Fighter's Action Surge, or the Haste spell.

The Dual Wielder feat works like this, at least by  consensus understanding of RAW:
1. You are wielding a Shortsword with Vex and a Scimitar with Nick. You take your Attack action to make an attack with your Shortsword, then an additional attack with the Scimitar because of the Nick property. Because both weapons have the Light property, you can then spend a Bonus Action to make an additional attack with either weapon, as either one qualifies as the triggering weapon for the Dual Wielder feat.

2. You are wielding a Longsword with Sap and a Scimitar with Nick. You take your Attack action to make an attack with your Scimitar. You cannot trigger the Nick property because you aren't making an extra attack with the Light property. However, because of the Scimitar's Light property, Dual Wielder allows you to spend a Bonus Action to attack with the Longsword, as it lacks the Two-Handed property.

3. You are wielding a Longsword with Sap and a Scimitar with Nick. You take your Attack action to make an attack with your Longsword. Because the Longsword lacks the Light property, you do not get to make any kind of follow-up attack, and everyone in your party makes fun of you for wasting your turn.

I personally believe RAI is that #2 should be the case regardless of whether you have a weapon with the Nick property, but that doesn't seem to be the shared consensus.

IF you had Action Surge or Haste cast on you, you COULD have a Shortsword and Scimitar equipped, attack with both and then stow them as part of your first Attack action, then on your second Attack action, draw a Longsword, wield it in two hands, attack with it, then use your Bonus Action to attack with the Longsword. Nothing in the feat says you can't make the Bonus Action attack two-handed, just that you can't make it with a weapon that has the Two-Handed property. The Longsword, which has the Versatile property, fits the bill. Your follow-up turns would get complicated, however, as you'd have to swap back and forth between them constantly to successfully continue. And your party would probably still make fun of you.

Also, yes, if you have the Two-Weapon Fighting feat, you can add your ability score to damage for all these attacks.

The shenanigans I was talking about are mostly misinterpretations of the draw/stow rules trying to claim that you get 2 draws and 2 stows with dual wielder (and at least one with the regular attack) that gives you enough to consistently draw both light weapons, do your attack action vex and Nick attacks with them then stow one of them, draw your longsword (etc) do the dual wielding bonus action attack with it then stow it and redraw the stowed light weapons. And repeat ad blockers infinitum during combat. I’m with you that that sort of misread based action should not be allowed. IMNSHO, dual wielding is really meant to grant light weapon wielders a third attack so their damage can keep up with great weapon fighters.

2014 Dual Wielding basically said you could wield two non-light weapons when two-weapon fighting, and I think the 2024 rule was trying to do the same thing, but they fumbled the wording. In all fairness, I think that's actually kind of a good thing, because the 2014 Dual Wielding feat was traaaaaaaash...

Whil I can - maybe - see wielding dual rapiers as actually working dual wielding longswords/katanas/ battle axes/etc is really pushing things. The problem is it doesn’t work in reality  even if it’s fun fantasy. I know we see things like Ashoka wielding dual light sabers and earth Maul wielding his version but we have keep n mind that these are carefully choreographed fantasy fights. Yes dand d are also fantasy fights but there are differences that need to be looked at and understood. All weapons are. ExTensions of the body ( mostly arms and hands) like the limbs they act as both offense and defense protecting areas of the body. Wing chun’s concept of defending the center line is a good place to start. Short light weapons provide both speed of reflex and the ability to cover one side of the body along with the centerline with each hand. Because the weapons are ( relatively) short they don’t cross not the other weapon’s area (much) interfering with that weapon’s defensive abilities. Longer, larger weapons extend further across the center line making portions of the other weapon’s defensive area unreachable until the crossing weapon leaves the area. His leaves you open to attacks in those undefended areas. Two short weapons or a long and a short weapon ( including shields for this) don’t interfere in this way because the short weapon’s arc can typically fit inside the arc of the long weapon while the longer weapon’s s arcs typically move outside the range of the shorter weapon leaving both able to attack and defend their entire areas at once. Try taking two broom sticks (or meter sticks) and trying to “sword fight” and see how often you smack either your own hands and arms or your two “swords” into each other. Then cut one down to half size and try it again and see what I mean about the differences. Two light weapons work well together, a light and a regular weapon work well together, a regular weapon and shield work well together but 2 long weapons mostly just get in each other’s way unless carefully choreographed again San eaqually choreographed foe. BTDT.