As suspected the press made a mistake in reportage. The Moravian was not P&O, it was White Star. Therefore Polly met Arthur on her passage between Sydney and London when he was working as a steward and they started an affair and she became pregnant. This would have happened by Dec 1900 and she obviously knew and told him when he returned from further runs in early Jan 1901, and they had moved in together at Plumstead by early Feb.
Also as suspected it is likely the loss of his job had something to do with this as well as the fact he moved out of his parents' home.
It is highly likely that John Grant retells a sanitized version of the story as more was known at the time than he lets on. He would have known that Polly was not some random woman who fell in the water and his son just happened to be strolling by and tried to come to the rescue but he decided to influence the story to make it more palatable.
I am not sure how accessible London papers were in Australia and New Zealand so how quick news was able to be conveyed. Telephones existed but were not common at this time, the first trans-Atlantic call took until 1927 but perhaps telegrams were popular by this time. Mary's sister in law Alice Tyree was still keeping a diary and writing letters and cards on their 1905 journey to London, published in 1996 by Arbour Press as 'A Budget of News.' She does not mention anything about Polly in it although, I will read it again to make sure I didn't miss anything.
I digress, Alfred Tyree, Alice's husband and Polly's brother, may have asked the manager of the London office to keep an eye on the press for any articles so they were informed (on how to not attract any attention to themselves).
Kentish Independent 4th May 1901
A Woolwich Mystery
The bodies of a Man and a Woman were found in the Thames last week, and at the inquest a verdict of "Found Drowned' was returned. On Saturday the deceased man was identified as Arthur E Grant, a Steward formerly employed on the P&O Liner Moravian. He and a young woman from Australia, whose name was Mary Tyrel, had been living together at the house of Mrs Harden, 28 Westdale Road, Plumstead and the woman was expecting to become a mother
ADDED
They are not at 28 Westdale Road on the 1901 census