I'm working on this - I have information on at least 4 male Uffmanns that emigrated to the USA, for one the connection to my family tree is only a vague guess based on his father's name and date of birth (I guessed that the father was the same person that his mother was married to).
I was wondering whether your William H. Uffmann was one of the twins that emigrated around that time, but too much of your information doesn't fit in. I have William H. Uffmann, born 20th April 1846 in Halle, Westphalia, who married Loisa Brinkman 1870 in Missouri - and for him, I have 2 children without any information about d.o.b. or names - and one Henry (Heinrich) Albert Uffmann, born 1st March 1874, as well as Emma Marie Christina Uffmann, born 25th Sep. 1884.
This William H. is about 7 years older than yours should be, lived in the wrong area and was married to the wrong woman - but I'll keep searching - I'll update you when I find anything.