In follow-up on what Danielle said, the change was made after Immigrants anglicized the spelling to better approximate the pronounciation of the name.
My 3rd-greatgrandfather and family immigrated to the USA in 1852. I found variations in census records fromKühn, Kiehn, Kihn, Kuehn, etc. as the census taker tried to enter the name from the spoken name.The cemetery tombstones of his sons have a combination of spellings with and without the umlaut -Kühn and Kuehn.
An anecdotal story recounted to my mother while she worked at a nursing home in the 1960-70s was that the change was made in the late 1800's because at the time they couldn't represent the umlaut on a telegraph message.Not sure how to prove that, but even in today's internet environment you often see a substitution of 'ue' for 'ü' on internet addresses.