I checked the Pulaski County (IL) cemetery books for you, and none of the men are recorded as being buried there. I don't own the Alexander County book (isn't for sale), but our local library has a copy. Should I find any of them I'll let you know.
I always try and check the whole 2 counties when someone is supposed/may be buried in the Mound City National Cemetery. Reason for this is that the doctors who treated the men (one of them at least) was the same one who oversaw the beginnings of the National Cemetery there; and his figures of total dead don't match up with the number that they use now as listed as unknown. The Confederates were originally buried on the riverbank at Cairo, IL, while many of the Union men were originally buried at a small cemetery in Pulaski County (then a field). These men died either on the way to the hospital, or after they arrived there.
When burying your relative at Columbus, often if the grave was marked at all, it was with a wooden plank--long since gone.
A newspaper article that mentions the sick and wounded from Pittsburg Landing, (about 200 or so named) who were on their way to Keokuk, IA; also mentions dropping off one or two on the way at St. Louis for burial. Perhaps your relative made it to St. Louis in this manner. The hospital boat was the Lousiana. Several were from IA regiments, but none of yours were named.
I did find one thing odd about the 14th Iowa though. There was a woman who impersonated a soldier of the 14th who committed suicide at Cairo. Didn't give her name, nor the soldier she was impersonating though.