Written somewhere round 1850 -or a little earlier- by Richard Monckton Milnes (words) and James Hine (music). It seems to have started as a poem by RMM (Baron Houghton, 1809-1885); it was also set later on (c.1875) by a Mrs Mackinlay.

The words were quite widely published on broadsides; see for examples  Bodleian Library Broadside Ballads:

As I Wander'd by the Brookside

Undated sheet music published by Ditson in America can be seen at the Lester Levy Sheet Music Collection:

I Wandered By the Brook-side. A Ballad.

I Wandered by the Brookside. Ballad

The song is number 2418 in the Roud Folk Song Index; it hasn't been found often in oral tradition; beside the text Alfred Williams got from Miss Leah Serman at Stanton Harcourt, Oxfordshire, Walter Pardon knew it and it is in the Copper Family repertoire. Helen Creighton found it once in Nova Scotia.

I don't know whether Barbara Berry set it to an existing tune or whether she made up a new one; presumably that information is in her book Down the Green Groves: Songs from Oxfordshire, Collected by Alfred Williams (1877–1930) (Kirtlington: Pedlar Music, 1989).