Now that you have made a few posts I can send you my direct e-mail by Personal Message.
I am now almost certain that Alexander Nic(h)olson, seaman, born 1825, married Mary Muir in 1847, had two children, Robert and Elizabeth, and died between 1851 and 1854, so I am not surprised that none of the Australian deaths matches him.
Their great-uncle Alexander Rhind, who died in 1841, bequeathed "To Robert Nicolson William Nicolson and Alexander Nicolson, all children of the deceased Robert Nicolson in Leven, the sum of Ten pounds Sterling each". This is in a codicil dated 29 February 1841, so it is reasonable to suppose that all three were living in 1841. There was no mention of Jean, Margaret or Thomas.
I have found Robert in the 1841 census and I think Alexander in the 1851. I have not yet found William in 1841.
If that image is the part of the bible with the date of death of Thomas, I cannot see it as 1883. I can only read it as April 1831.
The Fife Deaths Index records a death on 5 and burial on 8 April 1831 of Thomas Nicolson in Scoonie. The source of this information is the Scoonie parish register.
There are only two deaths of Thomas Nic(h)olsons in 1883 in Scotland, both infants.
Also, Thomas was not with his grandmother, mother and brother Robert in the 1841 census, and there is no Thomas Nic(h)olson in the census in Scotland who could possibly be him.
So if Thomas is not the one who died in Scoonie in April 1831, he must have managed to fall out with his great-uncle before the age of 12, leave (or be sent away from) home by the same age, evade the census enumerators and eventually die somewhere other than Scotland.
Robert became a tailor, and opened a shop in Dundee in 1841. However the shop is not listed in any of the trades directories, and I have yet to find Robert in the 1851 census, so I suppose that the shop was not a success and that Robert emigrated after the deaths of his mother in 1843 and grandmother in 1845. As he was the eldest son, I surmise that it would have been Robert who took the bible with him when he emigrated.
No idea what happened to William.