What I've got so far is as follows, with ? where I can't get a word and (?) where I'm not sure.

When I were small at me grandad's knee
He'd sing his old songs and his yarns to me
Of sailing ships and years long past
And serving time before the mast.
Sometimes a faraway look would come
And I knew I'd be hearing a special one.
Of every ship on every sea his favourite was the 'Wavertree'.

Running the easting down, running the easting down.
Proud to be a signed(?) AB on a Liverpool ship, the 'Wavertree'
Bowling along in a westerly, running the easting down.

A full-rigged ship two thousand ton
Southampton built for the Calcutta run.
When demand for jute began to fade
She entered the Australia trade.
In later years her ? crew ? a dirty cargo off Peru.
When she sailed ? eventually, a damn near starving(?) 'Wavertree'.

Running the easting down etc

For 25 years she played her part
'Til a Cape Horn snorter broke her (heart/mast?)
Almost forgotten, tired and worn
An old windjammer ? before.
If grandad were alive today he'd not believe it if you'd say
A century on a New York quay would berth his favourite 'Wavertree'.

Running the easting down etc.

All clarifications and corrections welcome.