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 probably around Woolwich somewhere by early Feb. They were not at Plumstead until early April as the census end Mar 1901 shows they are not residing there.
Also as suspected it is likely the loss of his job had something to do with this as well as the fact Arthur 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).
Just adding this from the other thread
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
Kentish Mail & Greenwich and Deptford Observer - May 3rd 1901 had this added
They left the house on Monday April 22nd, and after leaving the woman came back and threw her arms around Mrs Harden's neck and kissed her saying I regret that I have given you trouble you are a good woman. On the following day their bodies were found in the river. It has been stated they had become reduced almost to a state of destitution
Also in Lloyds Weekly Newspaper 28th April