Auscab; Oyster Bay Pine grows in the low rainfall areas of the east coast of Tasmania. It was fairly popular in colonial cottage furniture, both as solid timber and veneer – a mate has a shack near Triabunna that has Oyster Bay pine flooring and roof framing (probably built around 1850 as a shepherds cottage) – but I have never seen it in anything made since ~1900. That is probably when they ran out of mature trees. Remaining “trees” are more like big shrubs, and would yield minimal usable timber. I have never used it partly because the timber has never been available.

I googled and found that the latin name is Callitris rhomboidea and that it is a “native cypress”.
Oyster Bay Pine and South Esk Pine Woodland and Forest | Department of Natural Resources and Environment Tasmania

A bit more googling; apparently Callitris rhomboidea is also known as Port Jackson Pine – I thought OBP was indigenous to Tasmania, only, like Huon Pine, Celery Top Pine and King Billy Pine. We learn!
Australian Conifers

Keith Bootle calls it “dune cypress” and essentially dismisses it as being of “limited occurrence … but … similar to white cypress.” (p.271)

In your photo the marked marquetry pieces certainly look like Oyster Bay Pine, but I cannot be sure. Fascinating back story to that marquetry box at TMAG; Rick Reynolds, same family, was a prominent antique dealer/collector in Hobart in the era that you mention.