Thanks to Malcolm Douglas for the John Adams lead. Johnny gave me Keith Kendrick's email address who gave me this info:

"I have every reason to believe that it was not a 'maritime' song - as such - but a novelty item used in the English Music Halls in the 1800's, but Johnny found it in a Children's Radio broadcast support publication called 'Singing Together for Schools' in 1970.

It seems there are several interpretations of the term "rouse". In this case, I believe it refers to "heaving on a line" in order to "raise the sail". I believe it is itself a well - worn corruption or misuse (By the maritime fraternity itself) of the original meaning to "rouse" the sailors from their beds as in "rise", "arise",..."raise"? I'm afraid that's the best I can do, hope it helps. If you check through the text you wrote - I have, with helpful intention, entered a few corrections.
Anchors away
Keith

THE EAST INDIA MAN

It's many's the time I've sung this song when the wind's been blowing of a gale,
Hoisting up a yard or, shaking out a reef or rousing home the sheets of a sail.
I've shipped on board of a man-o-war, in the merchant service too,
And I've fought for me king and me country while I've sailed on the ocean blue.

Illy ally illy ally oh, cheerily boys cheerily
Bend your backs and give a pull
Cheerily I say I say
With a long pull and a strong pull
We'll haul away together boys
Belay every inch of that, belay boys, belay.

On a bright May day, we sailed away, on a big East India ship,
Though it's many, many years ago me boys, I'll not forget that trip.
We said goodbye to Portsmouth Docks, to Susan, Kate and Jane,
But we hadn'a been a sailing an hour or more when we joined in the old refrain...

Illy ally illy ally oh...

And now I'm off home from sea again going to put down me roots onshore,
Have a bit of a spell with me longhaired gal she's the one I do adore.
But there's no doubt should a war break out and seamen be required again,
Well I'll join with me crew me duty for to do and I'll join in the old refrain...

Illy ally illy ally oh...