They take normal damage and 1.5x/2x damage respectively.
Let's say Alex is the one who cast Warding Bond and some third party wizard cast Tether Essence.
The initial damage is D
Alex is hit first
If Alex is hit first Alex takes D damage and then Bailey takes D/2 damage due to Tether Essence and the resistance granted by Warding Bond.
Alex then takes damage equal to what Bailey took (D/2) due to Warding Bond
This again triggers Tether Essence causing Bailey to take (D/2)/2 damage.
Functionally this sets up a geometric series where D/2 is the first value and each subsequent value is half the former. The infinite sum of this series is (D/2)/(1-1/2) = D so for Bailey the effects of Warding Bond and Tether Essence cancel out except for the AC and saving throw buffs.
Alex will take damage equal to what Bailey will take (D) plus the initial damage causing Alex to take a total of 2D damage.
Bailey is hit first
If Bailey is hit first Bailey faces the same calculation and so takes D damage
Alex takes damage equal to what Bailey takes but also additional damage equal to the initial amount of damage Bailey took due to Tether Essence triggering alongside Warding Bond. As such Alex will take 3D/2 damage or 1.5D
In practice there will be some slight variation from these values due to rounding. For example if Bailey is hit for 23 damage that is halved to 11.5 which is rounded to 11. Bailey ends up taking 11 + 5 + 2 + 1 = 19 or rather D – 4. Alex would take 11 + 11 + 5 + 2 + 1 = 30 which is 4.5 less than 23 * 1.5
The reduction in damage increases at higher damage amounts but decreases in the percentage reduction. If you want an easy to use formula you could reduce the damage take to both parties by 2 for damage less than 25, 3 for 25-74, and 4 for 75-124, and 5 for any damage 125+. If you want the precise RAW value however it might be simplest to run the damage manually. For most damage values you will encounter this will be less than 5 iterations.
Why does Warding Bond trigger Tether Essence?
The spell descriptions indicate they do
There is nothing to suggest that damage resulting from Warding Bond does not trigger Tether Essence.
The wording of Warding Bond is:
Also, each time it takes damage, you take the same amount of damage.
While Tether Essence states:
When damage is dealt to one of them, the same damage is dealt to the other one.
Since both Warding Bond and Tether Essence are activated by the other party taking damage, and both deal damage when triggered, they will be triggered repeatedly until the damage done is 0.
There is the possibility that there is a difference between "taking damage" and "being dealt damage". Under this interpretation Warding Bond would not trigger Tether Essence because although the caster "took" equal damage they were not "dealt" damage. I have assumed "taken damage" and "damage dealt" are the same thing for my answer but as you noted in your answer it is not entirely clear.
Why wouldn't Tether Essence trigger itself?
Strictly speaking there is nothing in the wording of the spell or game rules to suggest damage resulting from Tether Essence does not trigger Tether Essence. Common sense, however, suggests that Tether Essence was not intended to do infinite damage so it cannot trigger itself.
This conclusion is not strictly RAW but rather gameplay necessity. This uses a similar justification as ruling Actions can be taken outside of initiative. It might not be stated in RAW but common sense and gameplay require it.
Is the same true of Warding Bond?
Warding Bond triggering Tether Essence does not produce the same gameplay issues as Tether Essence triggering itself.
For one Warding Bond exclusively applies to willing creatures thus, even if it were to do infinite damage, it cannot be easily exploited.
Additionally we can see that Warding Bond and Tether Essence do not create an infinite damage loop removing the gameplay concern.
Since excluding Warding Bond from triggering Tether Essence isn't required to facilitate gameplay and their wordings do not exclude each other there is no reason to impose such a restriction.
- The spell descriptions indicate that they can trigger each other
- No other rule prevents them from triggering each other
- There is no gameplay imperative to disallow them from triggering each other.