The spelling of the English language was not standardised until the late 19th century. It therefore comes of no surprise to find the standard of spelling of surnames even in official documents is whimsical particularly as regards census returns.
The majority of the population if they went to school, left at twelve years of age. The literacy rate was not good. A person spoke his name to an official who wrote down what he heard. The subject was not in a position to correct the spelling.
It is not uncommon to find related families to have differently spelt surnames. In fact in the UK EATLY and EATLEY appear to be interchangeable and in at least one case where the spelling was in doubt both variations were entered.(The marriage of EMMA in 1895)
Even if the surname was correctly recorded at the time, there is no guarantee that subsequent transcribers attempting to decipher the original entry got it right. For example in the UK the 1881 Census has the mistranscriptions EATTY & EATHY, and four families in Glamorgan, Wales who are actually HEATLEY are listed as EATLEY.
In the USA although there were EATLEYs before 1880 none are listed in the Census of that year. Either the surname became extinct or it was grossly mispelled in the original Census documents.
The variations of spelling so far encountered in order of frequency are, EATLY, EATLEY, EATELY, EITLY, EYAETLEY. There is the occasional ETLY but I wonder if this is related as it probably be pronounced with a short E. This is a line of enquiry to be explored. I also looked at EATLE but found one of the variations of this name amongst members of the same family was EATELL, I thought this unlikely to be a version of EATLY.