Dr. Thomas Braidwood Wilson of the Royal Geographical Society kept a daily log published in 1835 as Narrative of a Voyage Round the World; Comprehending an Account of the Wreck of the Ship Governor Ready in Torres Straits, which together with other first-hand accounts prove that the story in the OP is largely false:
The Mermaid wrecked 13 June 1829. "The crew took to the whale-boat, and, after three days, were picked up by the schooner Admiral Gifford, and shortly afterwards sent on board the Swiftsure; which ship was, on July 5th, wrecked off Cape Sidmouth" (quoting from page 83).
The Mermaid and Swiftsure crews were brought to land at Raffles Bay by the Resource 22 July 1829.
The Amity was in port at Raffles Bay when the Resource arrived and had been expecting the Mermaid, but the Resource came instead.
A detailed first-hand account of the wreck of the Governor Ready is also given, but this was an earlier event.
The Governor Ready was irreparably damaged at 2:45 PM 18 May 1829 (see page 12). The crew travels in three of their smaller boats (skiff, jolly-boat, long-boat) until they meet the docked Amity on 02 June 1829 (see pages 45-49). At this point the crew of the Governor Ready separates, Captain Young going to Batavia on the long boat, and Dr. Wilson going to Raffles Bay on the Amity. The Amity reaches Raffles Bay 2PM 31 June 1829 (see page 71).
Given that the author Dr. Wilson was actually aboard the Governor Ready when the wreck occurred, and personally met with the master of the Mermaid when the Mermaid crew arrived on the Resource, the portion of the story in the question concerning the Governor Ready is inaccurate.
As for the Comet, as reported in 19 October 1829 London Times, explaining that the second officer and a seaman of the Comet arrived in Cowes on 17 October (on the Maria having departed Batavia 30 June) to account for the wreck of the Comet to her owners and:
The 2nd officer stated that the Comet sailed from Sydney for Batavia on the 13th of April, made Torres Strait on the 5th of May. On the 6th at 4 a. m., thick weather, struck on a reef ... and was a complete wreck, the surf running so high that no boat could be put out obliged the crew to remain three days exposed ...subsisting on raw beef; at the expiration of that time succeeded ...steered north for Murray's Island, at 4PM saw it, and a brig at anchor; at 6 got on board, which proved to be the Fairfield, of Aberdeen, Captain Both, who recieved us politely, and attended to our immediate wants
As for the Jupiter, in the same article the 2nd officer continues:
on the 14th of May the Fairfield spoke the ship Jupiter...which had struck on Boot reef...she was proceeding to Calcutta for repairs
The 2nd officer further states:
Prior to the Maria's leaving Batavia, Captain Young, formerly of the ship Governor Reading [sic], of London, had arrived there with part of his crew, that ship having been unfortunately wrecked on a shoal in the vicinity of Murray's Island
So the order of events is:
6 May 1829: Comet wrecks (crew rescued by Fairfield)
14 May 1829: Jupiter is proceeding to Calcutta for repairs
18 May 1829: Governor Ready wrecks (crew eventually reaches the Amity 02 June)
13 June 1829: Mermaid wrecks (crew eventually rescued by the Admiral Gifford, transferred to Swiftsure)
5 July 1829: Swiftsure wrecks (crew rescued by Resource)
22 July 1829: Resource brings crews of Mermaid and Swiftsure to port at Raffles Bay where the Amity is expecting the Mermaid.
From this timeline is can be seen that the story in the OP is very inaccurate.
That the Mermaid and Swiftsure crews were transferred to the Resource rather than the Governor Ready is further corroborated by a letter from Samuel Nolbrow, the captain of the Mermaid published in the 30 October 1829 Colonial Times article, which further confirms:
Governor Ready wrecked 3PM on 18 May 1829.
Mermaid was abandoned 8PM 13 June 1829
Swiftsure was lost 4 July 1829 and crew was rescued by Resource.
Date of Comet wreck is not stated, but article says crew was rescued by the Fairfield.
As further confirmation see the first hand account of Collet Barker, commander of the Raffles Bay settlement publishied as Commandant of Solitude: The Journals of Captain Collet Barker, 1828-1831, which explains that Dr. Wilson first wrecked on the Governor Reidy [sic], and came to Raffles Bay on the Amity. Then on 22 July the Resource arrived with the crews of the Mermaid and Swiftsure.
Additionally, published in Historical Records of Australia: Despatches and papers relating to the settlement of the states, Australia Parliament Joint Library Committee, at page 835, is a 22 August 1829 letter from Collet Barker:
...loss of the Mermaid, which I regret to state was wrecked in Torres Strait on the 13 June. Her crew were all saved and were brought here by the Resource, the Master of which wished to have left them in consequence of being short of provisions and having with him the crew of another wrecked vessel, the Swiftsure. He consented however to take them except the Master, Mr. Nolbrow, on my supplying them with thirty two days provisions. I ordered Mr. Nolbrow on board the Amity and receiving an urgent request from Mr. Hastings,
Chief Mate, and Thos. Long, Seaman, to be allowed to go to Sydney, I also ordered them a passage in the same vessel
Also, the Jupiter arrived in Calcutta 5 July 1829, as recorded at page 32 of The Asiatic Journal and Monthly Register for British and Foreign India, China and Australasia, vol. 1, January-April 1830 and was not unrepairable but instead was in service in the Persian Gulf by May of 1830 (Calcutta Magazine and Monthly Register, October 1830, page 87).
Also in the above volume of Asiatic Journal:
The Swiftsure, Johnson, of London, from New South Wales, was wrecked 4th July Torres Straits. The crew and passengers were saved by the Resource, Smith, and had arrived at the Mauritius 17th Aug., with part of the cargo, stores &c.
So in summary the first hand accounts of Dr. Wilson, captain Nolbrow, and Collet Barker all show that the only true parts of the story are that the Mermaid wrecked, the crew was rescued by the Swiftsure which also wrecked and both crews were rescued. It is also true that the Governor Ready and Comet wrecked in the same time period, but these were earlier events, and the Mermaid crew was not rescued by these ships.
The inaccurate tale in the OP originates from a 26 November 1829 story in the Sydney Gazette, which could easily be mistaken for the truth, but for the existence of Mermaid Captain Nolbrow's 30 October 1829 letter published in the Colonial Times, the day-by-day accounts of Dr. Wilson and Collet Barker, and the account of 2nd officer of the Comet in the 19 October 1829 London Times.
The evidence agrees with the conclusion of maritime historian Charles Bateson, as stated in his book Australian Shipwrecks: 1622-1850 at page 87:
...both crews were rescued by the Resource which landed them at Port Raffles...statement that crews of the Mermaid and Swiftsure were taken aboard the Governor Ready and after the latter's wreck all three crews were wrecked in the Comet and finally, along with the Comet's crew, in the Jupiter, is ridiculous