The poem by Milnes is the oldest version known, so it should be credited to him. It is interesting to note that both old sheet music editions have the four verses unchanged.

Are the tunes used by Fisher and Cassidy similar to the old tunes by Hines and Schlesinger?

The Eva Cassidy rewrite with just verses one and four of the poem is posted below for completeness, but I think the third verse as included by Singing Referee adds a bit of drama.

I WANDERED BY A BROOKSIDE
(Eva Cassidy version)

I wandered by a brookside,
I wandered by a mill;
I could not hear the water,
The murmuring it was still.
Not a sound of any grasshopper
Nor the chirp of any bird,
But the beating of my own heart
Was the only sound I heard.

The beating of my own heart
Was the only sound I heard.

Then silent tears fast flowing
When someone stood beside,
A hand upon my shoulder
I knew the touch was kind.
He drew me near and nearer,
We neither spoke one word.
But the beating of our own two hearts
Was the only sound I heard.

The beating of our own two hearts
Was the only sound I heard.

Stuart Lowe has tabbed the Cassidy song: Brookside tabs