Ah, stop press...

Sharp does make several references to Child in 'Some Conclusions', such as:

'The extent and character of these variations may be studied with profit in the late Professor Child's English and Scottish Popular Ballads, where many of the ballads, which have in recent years
been collected in Great Britain, may be seen and compared with their European analogues. Indeed, as Mr. Andrew Lang has remarked: "It is unnecessary to indicate more than one authority on the subject of ballads. Professor Child of Harvard, has collected all known ballads, with all accessible variants, and has illustrated them with an extraordinary wealth of knowledge of many literatures.'

I'm afraid I don't have many early American folk song books on my shelves, so I can't say who was the first to adopt the 'Child and Other' ordering. I have Belden's 1912 review of the literature, but most of the collectors were publishing in JAF at the time, as far as I can see, and I'm not sure when the first books appeared. Wilgus would tell us, but I don't have that, and I know that Wyman and Brockway's Kentucky collection didn't place Child at the front. Is it even possible that Sharp and Campbell were the first to adopt the practice?