You can make unarmed strikes while Wild Shaped, but they do normal unarmed strike damage.
From the errata for the PHB:
Melee Attacks (p. 195). 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.”
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:
The most common actions that a monster will take in combat are melee and ranged attacks. These can be spell attacks or weapon attacks, where the "weapon" might be a manufactured item or a natural weapon, such as a claw or tail spike.
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.
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
Technically, they don't interact at all
While shifted you may make an unarmed strike as a bonus action (and only a bonus action) specifically with your fangs.
The Alter Self spell allows you to grow fangs etc with which you may make an unarmed strike, though not as a bonus action (unless you have some general ability that allows you to make any unarmed strikes as a bonus action). You would need to use your standard action to attack and these fangs are magic and have a +1 bonus.
Probably easiest to imagine - just to get the picture - that you created a different natural weapon with Alter Self, such as claws:
You would be able to use your standard action to make one (or more with Extra Attacks) unarmed strike with your claws. This attack is magic and has a +1 bonus.
Then you could use your bonus action to make an attack with your fangs. This attack would not be magic or have a bonus, as its just your standard shifted ability.
So...going back to the situation where you used Alter Self to grow fangs: You basically now have two sets of fangs. One magical with which you can use your action to attack and one non-magical with which you use your bonus action to attack!
However
Narratively, this is all very odd. As a DM, I would probably just rule that Alter Self allows you to 'grow' your standard fangs and they become magical (and gain a +1 bonus). This seems to make sense and probably wouldn't unbalance anything.
(I might possibly rule that you could use them only as part of your action OR your bonus action but not both, but even then I doubt it would be too unbalancing since you would have to give up your standard weapon attack if you used your action to make a fang attack).