I received a reply from the poster on the other forum and he doesn't know anything about the origin of the song or a source for it. He did, however, send me the melody, which I give below; maybe it will jog someone's memory.

Mick

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X: 1
T:Up And Down
S:sent to MCP by David Lamb of Foxhole Grace Baptist Church, Paignton, Devon
N:Original source still unknown
M:6/8
L:1/8
K:C
G2 E G3|G2 E GAB|cdc G2 ^F|F6|
w:Up and down, up and down on the wat-ers of Riv-er Nile,
F2 D F3|F2 D F2 A|BcB ABA|E6|
w:Up and down, up and down the crad-le was rocked for a while.
G2 E G3|G2 E GAB|ccc B2 _B|A3-A
w:By and by, Mos-es cried, just as loud as all bab-ies can,_
AB|ccc c2 A|ccc c2 A|GAB d2 e|c6|]
w:Till a love-ly princ-ess took him for her own un-til he be-came a man.