Hello Keith and Anne Caroline
I found the BDM info above a little difficult to follow, but a very interesting find - especially as the rest of the family seem to be hiding. I think it comes from the AVRI and probably the Tasmanian archives. This is the link for the archiveshttp://www.archives.tas.gov.au/
and this for the indexeshttp://www.archives.tas.gov.au/nameindexes
Given the timeframe, if your William did end up in Hobart, there would be a better than 50/50 chance he was a convict. There were, however, from the archives site 17 William Carters who were sent to Tasmania
and age is not mentioned in the list! You could discount a few who came before 1830, but still many left. I believe it was not uncommon for wives with/without children to follow their husbands to Australia - but I doubt it was a very frequent event.
Apart from Lydia Carter (if she is yours) I can see none of the other children in the Family links data base (which is taken from the Tas BDM records to 1900). These are the same BDM records that are on the AVRI.
These are the Tasmanian records I haveMarriage
1846 Ref 2313 Lydia Tidmarsh Carter aged 21 M Lavington ROOPE Aged 25Children - All Births
George 6 Jan 1847 Ref 2187
Female 25 May 1848 Ref 1080
Female 9 July 1849 Ref 1678
Female 17 Jul 1850 Ref 2700
Male 27 Jul 1851 Ref 907
Male 22 Dec 1860 Ref 1861/4128
Raymond Hamilton 25 Oct 1857 Ref 1858/1310Deaths
Alice ROOPE age 11 8 Jan 1860 (presumably the lass born July 1849) ref 1866
Lydia Tidmarsh ROOPE age 40 8 Sep 1864 Ref 4689
Lavington ROOPE age 60 29 Jun 1881 ref 3197
NSW can be checked online at www.bdm.nsw.gov.au
- follow the family history links
The closest I found for a death of William Carter that suits your information were 2 from Victoria
Death: William Carter Father: Robert Mother Mary Evans Age 81 Birthplace UNK Year 1878 Ref 10035
Death: William Carter Parents Unknown Birthplace Unknown Age 77 Year 1871 Ref 5814
The first has the wrong mother (but is the right age), the second is a few years out on age. Informants often got both of these items incorrect - but neither of these may be your man.
Victorian death certificates can be downloaded - they cost $17.50 (abt 7 pounds) so expensive if not the correct one!
There are a number of links to Vic immigration & other information on the Victorian Links thread at the top of the board which may be useful. Some of the indexes available are herehttp://www.access.prov.vic.gov.au/public/PROVguides/PROVguide023/PROVguide023.jsp
The Tas Archives staff are very helpful. I don't know of alot of Tas immigration that is online, but the records are at the archives. If you browse the site it may tell you where to look - or if you can find nothing you can try an email to the archives.