Somewhere I read that the bows are the pieces of land edged by a meandering river.

I have a tape with this song being sung by Martin Carthy as posted above. (Dec 14 2002). I've tidied it up to suit myself. In some places, I think my memory has been more accurate.

Here's two royal sisters walking along
Hey the gay and the grinding.
Rock dove sing a mournful song
By the bonny bonny bows of London

And the eldest has pushed her sister in
Hey the gay and the grinding
Pushed her in to sink or swim
By the bonny bonny bows of London

And the eldest has pushed her sister down
Hey the gay and the grinding
Pushed her in and seen her drown
By the bonny bonny bows of London

Oh she's floated nigh and she's floated yon
Hey the gay and the grinding
Floats till she come to the miller's pond
By the bonny bonny bows of London

Next comes along the miller's young son
Hey the gay and the grinding
"Father dear here swims a swan."
By the bonny bonny bows of London

Fool with a fiddle comes riding by
By the bonny bonny bows of London
Pulls her to the bank to dry
By the bonny bonny bows of London

And he's made some strings of her long yellow hair
Hey the gay and the grinding
Pulled them taut and tuned them fair
By the bonny bonny bows of London

And he made some pegs from her long fingerbones
Hey the gay and the grinding
Made some pegs and made some drones.
By the bonny bonny bows of London

And he made a fiddle of her breastbone
Hey the gay and the grinding
So's 'twould would pierce a heart of stone
By the bonny bonny bows of London

And he's fiddled by night and he's fiddled by day
hey the gay...
only tune that the fiddle would play
Was the bonny bonny bows of London

So the fool's gone down to the king's high hall
Hey the gay and the grinding
Music, dancing, feasting and all
By the bonny bonny bows of London

And he laid this fiddle all on a stone
Hey the gay and the grinding
Fiddle began to sing all alone
By the bonny bonny bows of London

"Yonder stands the wand'ring fool"
Hey the gay and the grinding
"Yonder sits my sister cruel."
By the bonny bonny bows of London

"Yonder sit the queen and king."
Hey the gay and the grinding
How they'll weep at my burying."
By the bonny bonny bows of London (2x)