Can a Fighter with Two Weapon Fighting Style using a short sword and scimitar respectively get 3 attacks per round?
If the Fighter is at least level 5, and can therefore make two attacks with an Attack action, then yes, though that doesn't require the Nick property.
The Nick property lets you make an off-hand attack with a second weapon as part of the same action you use to make a normal attack with your main hand weapon, which would normally require a bonus action to do. It doesn't let you make more than one off-hand attack on the same turn.
A fighter needs to be at least level 4 if sticking strictly to Player's Handbook content.
The Nick Property does not change the number of attacks you can make in a round. It just makes the one attack you make from the Light Property part of the Attack Action. So you go from 1 attack as an Attack action and one attack as a Bonus action to 2 attacks as an Attack action. If you then have a way to attack again as a bonus action, such as with the Dual Wielder feat, you could make a third attack. There may be a way to do it sooner with Origin Feats and a Cantrip, but I can't think of any.
Two Weapon Fighting Style doesn't help in terms of the number of attacks; it just adds you attribute bonus to your damage.
As @SmiteMakesRight_3_5 said, before reaching level 5, you'd need the Dual Wielder feat to make three attacks by using the Bonus Action that Nick freed up.
I usually recommend this video. It explains well the interaction between the Light weapon property, Dual Wielder feat, Two-Weapon Fighting Style feat, and Nick Weapon Mastery:
A fun thing to note: if you are allowing the old races at your table using the backwards compatibility rules, a Thri-Kreen can use his extra hands to wield light weapons. His main hands would still be free to wield whatever, which means at level 5 you could have a character with an attack combo like this:
All of this can be done without stowing/drawing weapons, and you have the option of using two hands for the longsword attacks (versatile), or carrying a shield in your other main hand.
Can a Fighter with Two Weapon Fighting Style using a short sword and scimitar respectively get 3 attacks per round?
If the Fighter is at least level 5, and can therefore make two attacks with an Attack action, then yes, though that doesn't require the Nick property.
The Nick property lets you make an off-hand attack with a second weapon as part of the same action you use to make a normal attack with your main hand weapon, which would normally require a bonus action to do. It doesn't let you make more than one off-hand attack on the same turn.
pronouns: he/she/they
A fighter needs to be at least level 4 if sticking strictly to Player's Handbook content.
The Nick Property does not change the number of attacks you can make in a round. It just makes the one attack you make from the Light Property part of the Attack Action. So you go from 1 attack as an Attack action and one attack as a Bonus action to 2 attacks as an Attack action. If you then have a way to attack again as a bonus action, such as with the Dual Wielder feat, you could make a third attack. There may be a way to do it sooner with Origin Feats and a Cantrip, but I can't think of any.
Two Weapon Fighting Style doesn't help in terms of the number of attacks; it just adds you attribute bonus to your damage.
The Nick, Dual Wielding, and non-light weapons post may interest you.
How to add Tooltips.
As @SmiteMakesRight_3_5 said, before reaching level 5, you'd need the Dual Wielder feat to make three attacks by using the Bonus Action that Nick freed up.
I usually recommend this video. It explains well the interaction between the Light weapon property, Dual Wielder feat, Two-Weapon Fighting Style feat, and Nick Weapon Mastery:
Tried to do a TarodNet impression only for the real deal to show up with an audio / video presentation. 😉
Also, I don't know how you gather your references with the search tools here.
How to add Tooltips.
Hahaha 😅 I can send you a private message, my friend!
A fun thing to note: if you are allowing the old races at your table using the backwards compatibility rules, a Thri-Kreen can use his extra hands to wield light weapons. His main hands would still be free to wield whatever, which means at level 5 you could have a character with an attack combo like this:
1st Attack - Shortsword , Nick Attack - Scimitar, 2nd Attack, 2nd Attack - Longsword, Dual-Wielder Bonus Attack - Longsword
All of this can be done without stowing/drawing weapons, and you have the option of using two hands for the longsword attacks (versatile), or carrying a shield in your other main hand.
A human wielding a Shortsword and Scimitar coud also do the same with this feat.
1st Attack - Shortsword , Nick Attack - Scimitar, Unequip the two weapons as Free object Interaction
Equip two weapons as Attack action 2nd Attack - Longsword, Dual-Wielder Bonus Attack - Longsword