Well, of course it isn't in Child. Neither is 'There are Fairies at the Bottom of My Garden' or 'The Dong with the Luminous Nose'; and for much the same reasons. The melody doesn't need to sound 'celtic', whatever you understand by that, because the lyric is just a typical piece of bourgeois Victorian romanticism. The fact that it was written by somebody who happened to have been born in Scotland is completely irrelevant to its content, style and tone. As you presumably know, of Mrs Richardson's poems printed in Nimmo only one, 'Ah! Faded is that Lovely Bloom', even mentions a tune; and that states merely "Written to an Italian air".

It may well have been set by one or more people over the years, but I find no trace of any such setting. It would be in the standard parlour song style, I expect, if it exists; certainly not anything that would sound 'celtic' to the modern ear.