The British could have easily done this, but they'd have to get the ball rolling early on to avoid the White Australia scenario. My personal opinion, on the easiest and most plausible way to get this up and running? Have the British East India Company establish settlements in Northern Australia, perhaps at around the same time as their Malaccan Straits Settlements, which were established in 1826 to protect its trade route to China and to combat local piracy, and as Stirling's Swan River Colony, which was established in 1829. And ITTL, couldn't the Northern Australian Settlements become the British East India Company's primary penal settlements for Indian civilian and military prisoners, instead of Singapore, Malacca, Penang and Dinding? If they had, then these penal settlers alone would have been enough to give Northern Australia a population at least on a par with that of Western Australia into the 1850's.
And once the floodgates opened with the Great Uprising of 1857, with the vast numbers of surplus mutineers, rebels or recaptured prisoners, amounting to over 80,000, and with the penal settlements in Burma and the Straits refusing to take them, most of these would probably be shipped over to Northern Australia instead, since these penal colonies would have by far the most room for expansion, instead of all of them being shipped over to the relatively cramped, unprofitable and resource poor Andaman and Nicobar Islands- with Northern Australia's population easily surpassing those of both OTL's Western Australia and Queensland combined (and most likely encompassing the northern portions of both) by the early 1860s. And from there, the sky's the limit- North Australia's population ITTL would almost certainly exceed 5M, and could easily exceed 10M.