Bonzo - it looks like Thomas Whittaker came over to New South Wales, leaving on 25 May 1812 on 'Earl Spencer' for 7 years. Convicted at Chester so it looks like the correct person.
He was aged 44 but was a Joiner, according to that particular record. Earl Spencer going to New South Wales.
This is on the 'Find My Past' site -- but there is another image on that site that doesn't seem to open for me - if anyone else wants to try (From A-Z of record sets -- type in Australia Convict -- as you type a list will appear.)
The image I cannot open is for Thomas Whittaker 1812 Cheshire - Indents 1788-1842
(It might not say much but it would be good to know - I can't understand why I cannot open it).
In addition on the Ancestry site there is an 1828 Census Extract showing Thomas Whittaker, aged 60 - arrived 1813 on Earl Spencer. He is a Mathematics Teacher in 'Rd Cook Sydney' -- it is very difficult to read.
Unfortunately it is not a census as we know it - not split into family groups etc. However it tells us where he was and his occupation at that point. (He did well for himself didn't he!)
Bonzo -- you may find something in the Australian Newspapers - Trove website-- it's a free site and it's brilliant;
https://trove.nla.gov.au/newspaperIf you use the Advanced Search option you can better control your results.
As he was a teacher you may just find something about him.