Hello April
I hope you're still reading these boards as this is an old message.
I have a Bernasconi family in Australia - but I don't think it's the same one as yours as my John Bernasconi came from Milan, Italy and didn't have a son Thomas. But the listings I found for the pictureframe maker in Sydney I thought were my John but your information here would seem to rule that out.

So here is what I had on John B, the pictureframe maker and others (records available through Ancestry):

SANDS DIRECTORIES (of Sydney and New South Wales, Australia)
1905, p. 704
Bernasconi C. A., picture-frame maker, 72 Royal arcade, George st
Bernasconi E., 39 Union st, Paddington , 1903-1916

1898, p.618
Bernasconi J., 115 Eveleigh st, Redfern
Bernasconi John, pictureframe maker, 347 Pitt street

1889 p.479
Bernasconi Brothers, pictureframers, 3
Paddington st, Paddington
Bernasconi Ernest, pictureframe maker,
347 Pitt street

1886 p.419, to 1889
Bernasconi Brothers, picture-frame
makers, Paddington st, Paddington
Bernasconi, John, picture-frame maker,
347 Pitt st

I don;t have anything on Thomas Bernasconi though, sorry.
cheerio