Two weapon fighting with Nick Vex?

Can a Fighter with Two Weapon Fighting Style using a short sword and scimitar respectively get 3 attacks per round? 

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.

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: Why You Get 4 Attacks a Round by Level 5 in Dungeons & Dragons 2024 | D&D 2024 Rules Update

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.

Quote from SmiteMakesRight_3_5 >>

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.

 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