When Cowdery returned to Church membership in 1848 he spoke to an Iowa conference. His words there were recorded by Reuben Miller. Recall that on this occasion, he had possession of the brown stone that some say Joseph used to translate the Book of Mormon, yet Oliver did not mention that stone at all. He did not display it to the audience to prove his testimony.
Instead, he reaffirmed the translation by the Urim and Thummim.
“I wrote with my own pen the entire Book of Mormon (save a few pages) as it fell from the lips of the Prophet as he translated it by the gift and power of God by means of the Urim and Thummim, or as it is called by that book, holy interpreters. I beheld with my eyes and handled with my hands the gold plates from which it was translated. I also beheld the Interpreters. That book is true. Sidney Rigdon did not write it. Mr. Spaulding did not write it. I wrote it myself as it fell from the lips of the Prophet. It contains the everlasting gospel and came in fulfillment of the revelations of John where he says he seen an angel come with the everlasting gospel to preach to every nation tongue and people. It contains principles of salvation. And if you will walk by its light and obey its precepts you will be saved in the everlasting kingdom of God.
"Bro. Hyde has just said that it was all important that we keep in the true channel in order to avoid the sandbars. This is true, the channel is here, the priesthood is here. I was present with Joseph when an holy angel from God came down from heaven and conferred, or restored, the Aaronic priesthood. And said at the same time that it should remain upon the earth while the earth stands. I was also present with Joseph when the Melchizedek priesthood was conferred by the holy angels of God which we then confirmed on each other by the will and commandment of God.”
You can see the original Reuben Miller journal entries about Oliver's return to the Church here:
A good article about the translation that references the journal is here: