Most of the apocryphal gospels are dismissed by scholars as being too late to tell us anything useful about the historical Jesus, but the Gospel of Thomas may be an exception. When the Jesus Seminar (a group of around 200 scholars who study the historical Jesus) debated which sayings attributed to Jesus in the gospels were most likely authentic, they concluded that none of sayings in John's gospel were certainly authentic, but that three sayings in the Gospel of Thomas were unquestionably authentic.
This is the passage, from saying 13 in the Gospel of Thomas, that suggests Jesus used the divine name:
And he took him and withdrew and told him three things. When Thomas returned to his companions they asked him, "What did Jesus say to you?"
Thomas said to them, "If I tell you one of the things which he told me, you will pick up stones and throw them at me; a fire will come out of the stones and burn you up.
Given the sanction of stoning mentioned here, many scholars think the reference is to Jesus using the divine name, perhaps alluding to Exodus 3:14 where three words are used to explain the one name. So this may be a relatively clear case of the divine name on the lips of Jesus Christ from a relatively early source. But given it is found in an apocryphal gospel I doubt the Watchtower Society will be quoting it any time soon!