Hello Christy,
As you are probably aware birth certificates were not required until 1905. You've looked in the right place in the register of births; usually completed by the midwife and filed with the county. In 1898, the state legislature provided for central county records, requiring county clerks to keep separate birth and death registers. Since marriage was seen as a religious sacrament, the civil registration of marriages was not required in Utah until 1887.
I have several books on Sanpete county and one in particular for Mt. Pleasant. It was published in 1947 and belonged to my dad. It is about 340 pages with no index but I did look at each page for the last name of Oldham. There were a couple of snippets, but not much.
Lewis Oldham lost his life in a flood in June of 1918. He slipped and fell into a stream. His body was found some days later in the debris west of the city.
John Oldham traveled with the Marin & Willie handcart co and was part of the men and women who pioneered Mt. Pleasant.
On 27 May 1937 one of the most destructive storms in Sanpete history hit N. Sanpete Valley. The home of Charles Oldham was nearly destroyed by lightening.
This last citation came from a different book entitled "These Our Fathers" 1947.
I hope this is your family and that some of this information is of interest to you.
Linda Holmstrom