The primary sense of the Australian-English compound underground mutton is rabbit meat; this compound has also come to designate rabbits.

The two texts containing the earliest occurrences of underground mutton that I have found indicate that rabbit meat was consumed when butcher’s meat was too costly:

1-: From one of the unconnected paragraphs making up the column Here and There, published in the Murrumburrah Signal (Murrumburrah, New South Wales) of Saturday 8^th September 1900:

Meat has gone up a ½d in the lb in Murrumburrah and Harden, and milk has dropped 1d per quart. Some poor folk declare they will live on underground mutton (wild rabbit) until meat gets cheaper.

2-: From The Areas’ Express and Farmers’ Journal (Booyoolee, South Australia) of Friday 9^th October 1903:

Spalding.
October 5^th, 1903.
Something new in the line of strikes occurred here lately, much to the surprise of Messrs Naylon and Millington, two of the camps containing over 100 men struck against the price these butchers were charging for their meat. The men, after living on “underground mutton,” for a few days were eventually successful in bringing down the price one penny a lb., and now they are satisfied.

Likewise, the following is from District Correspondence: Campbell’s Creek, published in The Mount Alexander Mail (Castlemaine, Victoria) of Friday 6^th September 1912:

The abnormally high prices of the necessaries of life, coming on top of a bad season, are weighing very heavily on the average householder. Potatoes, onions and meat are twice the price of a few years ago. Taking advantage of a week’s respite from school, the meat bill is being lessened for the time by parties of lads out rabbiting, and underground mutton is a more regular dish at the table than usual.

In underground mutton, therefore, the second element, denoting a choice meat, was derisively substituted for the noun denoting the inferior meat that had to be eaten when butcher’s meat was too costly—underground mutton is thus comparable to terms such as:
– a poor man’s goose, denoting a casserole dish of liver and potatoes;
– weavers’ beef, denoting sprats;
– Welsh rabbit, denoting a dish consisting of melted cheese on hot buttered toast;
– Irish apricot, Irish grape, Irish apple, Irish lemon, Irish plum and Munster plum, denoting the potato;
– Bombay duck, denoting the bummalo fish.

However, in certain circumstances at least, rabbit meat added variety to the diet—as exemplified by the following from a description of the harvesters’ camp at Nagambie, by ‘W. M.’, published in The Herald (Melbourne, Victoria) of Thursday 31^st December 1903:

We get out the frying-pan from the tucker-bag. It is surprising how quickly a meal is cooked, and with what a keen relish a repast of steak and onions, bread and treacle, and tea is partaken of in the open. You live well in camp at a trifling cost. Provided with plenty of tea and sugar at the start, the victualling for half a week cost me less than sixpence a day, and I had as much as I could eat.
Yes, you live well on very little. The day I met Ginger he was eating what I thought was a chicken.
“What’s that you’ve got?” I asked.
“Underground mutton—bunny.”
“Buy it?”
“Buy it? H—, no. I carry three rabbit traps. I believe in a change of diet when I can get it.”

Similarly, the following is from District Correspondence: Campbell’s Creek, published in The Mount Alexander Mail (Castlemaine, Victoria) of Tuesday 15^th October 1907:

Rabbits are very numerous about here this season and are of great annoyance in keeping the crops and grass short in these dry times. It is no uncommon thing for parties going out for a change of diet in underground mutton to shoot rabbits from their vehicles, so daring have they become.

The following, from The Advertiser (Adelaide, South Australia) of Wednesday 14^th December 1904, contains an early occurrence of underground mutton in the sense of rabbits:

Melrose. December 10.—[…] Snakes are not very numerous this season, but iguanas are very plentiful, ranging in size from 1 ft. up to about 6 ft. in length. Those that infest the haunts of the underground mutton (rabbits) are not destroyed, but the hen-egg robbers are dispatched without mercy.

The National Museum of Australia explains:

In 1859 European wild rabbits were introduced into Australia so they could be hunted for sport. Within 50 years rabbits had spread across almost the entire continent, with devastating implications for Australia’s indigenous flora and fauna. The proliferation of rabbits was the fastest of an introduced mammal anywhere in the world.

About the proliferation of rabbits, I have found this interesting article, published in The Canowindra Star and Eugowra News (Canowindra, New South Wales) of Friday 11^th May 1906:

The “Rabbit Factory.”

Operations in connection with the “rabbit factory” mentioned in our last issue are to be commenced almost immediately—probably on Monday next—and it is anticipated that well within six weeks the process of converting “bunny” into a saleable article of commerce will be in full swing. The plant for the present we understand will not be an expensive one—probably of the value of between £300 and £400, but capable of dealing with a large number of rabbits daily. “Rabbit extract” for export and probably to be put up in small jars for retailing, for the present will be the only finished article manufactured locally, whilst the meat fibre will be forwarded to Sydney in bulk, there to be converted into dog biscuits, sold for fowl feed, or utilized in other ways. The skins, of course, will also be marketed, and the supply being expected to be large, doubtless will be exported direct to other markets. Messrs Clarke and Barnes, who are establishing the factory intend to run all the year round, and by reason of this fact should establish a large connection, being the handiest market for this and adjacent districts, which will warrant the duplication (which is provided for) of the plant. We are given to understand that “bunny” will be purchased at so much per 100lbs all the year round, so that anything in the form of a rabbit will be acceptable, and thus the “underground mutton” squatter will have a market for the small fry as well as the full grown. We wish the enterprising proprietors every success, and hope to see bunny prove that instead of being a pest without any good points, his virtues are far more numerous than his vices.