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The Unearthed Arcana's Oath of Redemption has the following features:

Aura of the Guardian

Starting at 7th level, you can shield your allies
from harm at the cost of your own health. When a creature within 10
feet of you takes damage, you can use your reaction to magically take
that damage, instead of that creature taking it. This feature doesn't
transfer any other effects that might accompany the damage, and this
damage can't be reduced in any way.

At 18th level, the range of this aura increases to 30 feet.

Emissary of Redemption

At 20th level, you become an avatar of peace,
which gives you the following benefits.

• You have resistance to all damage dealt by other creatures (their
attacks, spells, and other effects).

• Whenever a creature damages you, it takes radiant damage equal to
half the amount it dealt to you.

If you attack a creature, cast a spell on it, or deal damage to it by
any means but this feature, neither benefit works against that
creature until you finish a long rest.

Now obviously the damage couldn't be reduced by Emissary but I can't find if it should be able to reflect half of it.

On one hand, you are taking damage dealt by another creature, but on the other the creature isn't attacking you, you're simply taking the damage for someone else.

This works for the UA version of the Oath of Redemption

As you already quoted, in the UA version of the Oath of Redemption that you are using, by transferring damage using Aura of the Guardian, Emissary of Redemption should probably trigger, since you are taking damage "instead of the other creature".

However, keep in mind that you are using a playtest version of the Oath of Redemption. As usual, the UA includes at the beginning:

The material here is presented for playtesting and to spark your imagination. These game mechanics are in draft form, usable in your campaign but not refined by design iterations or full game development and editing. They aren’t officially part of the game and aren’t permitted in D&D Adventurers League events.As is typical in Unearthed Arcana, the options here haven’t been tuned for multiclassing.If we decide to make this material official, it will be refined based on your feedback, and then it will appear in a D&D book.

Emissary of Redemption was changed in Xanathar's Guide to everything

In this case, the Oath of Redemption has indeed later been published and the Emissary of Redemption feature was changed to only apply to attacks directed at you. It now reads (emphasis mine):

• Whenever a creature hits you with an attack, it takes radiant damage equal to half the damage you take from the attack.

Being hit with an attack has a specific meaning in D&D 5e that is not just taking damage. The mechanics for how an attack works state that:

When you make an attack, your attack roll determines whether the attack hits or misses. To make an attack roll, roll a d20 and add the appropriate modifiers. If the total of the roll plus modifiers equals or exceeds the target's Armor Class (AC), the attack hits

So first, an attack roll is made against a creature to determine if the attack hits. Then any effects for a hit or a miss happen, including damage. When you transfer damage using Emissary of Redemption, even if it is damage from an attack that hit another creature, the paladin was never actually hit with the attack. Therefore, Emissary of Redemption does not apply.