Myujth

In which yuga was Dattatreya born? Was it Dwapara yuga or Kali yuga?

According to Suta Samhita (of Skanda Purana), Lord Dattatreya was born in Satya Yuga (Krita Yuga).

This Kamakotimandali article mentions that:

There are other incidents described in the Puranas regarding
Dattatreya Avatara. The Parabrahman appeared as yogirAja in front of
Atri Maharshi who was performing penance on RkSha Parvata on kArtika
shuddha pourNami. He again appeared to Sri Atri on kArtika bahuLa
pADyami as Atrivarada. Pleased with the penance of the sage, he
granted (datta) himself to Atri and came to me known as Dattatreya. It
was at this stage that the trinity tested anasuya. From Suta Samhita,
one can know that Datta Avatara first occurred in Krta Yuga –
vaishAkha bahuLa dashami - Thursday – Revati Nakshatra mIna lagna. The
incident of Sri Dattatreya discoursing to Prahlada further proves this
fact.

It also gives additional details of birth like Janma Tithi, Lagna, Nakshtra etc. And, as regards the Manvantara, it is mentioned as Swayambhuva.

Thus, considering the present cycle of creation, the great manifestation of the Supreme as Dattatreya happened in swAyambhuva
manvantara. We have completed swAyambhuva, swArochiSha, uttama,
tAmasa, raivata and chAkShusha manvantaras and the current manvantara
is Vaivasvata. Every Manvatara approximately maps to about 3086 crore
years. Thus, from the beginning of the latest cycle of creation, we
have now seen 3086 X 6 = 18516 crore year

And, that he could not have been born in either Dwapara or Kali is evident from the fact that he taught Sri Vidya to Parashurama in Treta Yuga from which the "Parashurama Kalpa Sutra" was composed.