The rule for Two Weapon Fighting is as follows:
When you take the Attack action and attack with a light
melee weapon that you’re holding in one hand, you can
use a bonus action to attack with a different light melee
weapon that you’re holding in the other hand.
The errata for the PHB states that unarmed strikes are not weapons, so you can't use Two Weapon Fighting at all.
The rule on
unarmed strikes should read as follows:
“Instead of using a weapon to make a
melee weapon attack, you can use an
unarmed strike: a punch, kick, head-butt, or
similar forceful blow (none of which count
as weapons). On a hit, an unarmed strike
deals bludgeoning damage equal to 1 +
your Strength modifier. You are proficient
with your unarmed strikes.”
With your houserule, the DPR is huge
Natural Weapons are not Unarmed Strikes, making them so is a very unofficial, very unbalanced houserule.
Level selection
The best damage available in beast form is 24(4d8+6, Triceratops Gore) at +9 to hit, but you need 15 levels of Druid for it. To get the Extra Attack, you have to have 5 levels of Monk.
This is all of your 20 levels.
Starting Ability scores
for Human Variant with the Resilient(Con) for Concentration:
Str: 8
Dex: 16
Con: 14
Int: 10
Wis: 16
Cha: 10
ASI Monk 4: Sentinel, for out of turn DPR
ASI Druid 4: +2 Wis, for AC
ASI Druid 8: +2 Dex, for Flurry of Blows
ASI Druid 12: +2 Dex
Calculation
Assume AC 20 for the enemy, it is not unreasonable on level 20.
Official DPR:
+9 to hit, 2 x 24 (Gore) = 0.5 x 2 x 24 = 24
+11 to hit, 2 x (3.5+5) (Flurry of Blows) = 0.6 x 2 x 3.5 = 10.2
Alltoghether it is 34.2 vs AC20.
This is a very sudden jump however, at level 19 you either do not have Triceratops, or Extra Attack. Without Extra Attack you do 12 + 10.2 DPR, below a 17th level Monk.
Houserule DPR:
+9 to hit, 2 x 24 (Gore) = 0.5 x 2 x 24 = 24
+9 to hit, 2 x 24 (Gore as FoB) = 0.5 x 2 x 24 = 24
Alltoghether it is 48 vs AC20.
Extra Attack makes a much smaller difference now, as the FoB does the same damage as attacks done with your action. This is very unbalanced.
Compare it to 17 levels of Monk:
+11 to hit, 2 x (1d10+5) (basic attack) = 0.6 x 2 x 10.5 = 12.6
+11 to hit, 2 x (1d10+5) (Flurry of Blows) = 0.6 x 2 x 10.5 = 12.6
This all remains the same in the next 3 levels; 25.2 vs AC20.
Best Answer
You can make unarmed strikes while Wild Shaped, but they do normal unarmed strike damage.
From the errata for the PHB:
An unarmed strike is now defined extremely clearly. On the plus side, an unarmed strike can replace any attack with a weapon. From the Monster Manual, page 10:
So your unarmed strikes can definitely replace your natural attacks.
The downside to this is that your natural attacks definitely don't count as unarmed strikes, so you can only do your normal unarmed strike damage. This was also confirmed by Jeremy Crawford.