Certainly a Russian colony in Australia would not have been too implausible in Thande's Look to the West. It's much too late for that easy approach in his timeline now, though conceivably there could be Russian conquest or even acquisition by purchase. To get the whole of Australia, the Tsar would have to deal with the French, the English, and the associated but de facto independent Empire of North America. The way things are going there it might still be possible for the Tsar to be allied with one of these powers and take on the job of conquering the sections controlled by one or two of the others, then for Russian settlement to swamp the English and French speakers there.
And in this scenario, or the already lost alternate possibility of a Russian settlement from the get-go, it is entirely plausible the colony stays loyal to St. Petersburg/Moscow, whichever one is serving as the Russian capital at the time. (At this time in the timeline, both are nominally capitals; as Lithuania spins farther out of the Russian orbit, I guess Moscow would get a leg up, though there is something to be said for staying in Peter's window on the west to keep an eye on things. But nativist Russian nationalism is getting stronger, that would tend to vote for Moscow).
I doubt very much the Russians get control of even part of Australia in LttW, but you never know with Thande! (Can we blame him?

) They already have control of Japan, and the Philippines may fall to them yet. I believe the Portuguese are currently running things there, nominally on behalf of their puppetized Spain which has now fallen to the New Spanish Empire, so perhaps they've been handed back, perhaps the Portuguese now hold them openly in their own name or obstinately as an even more clearly puppetized last holding of the other Spanish dynasty. So yeah, the Philippines are up in the air and wars may be fought over their control, and the Russians might well get involved, which might have them colonizing Australia after all.
Taking a fresh timeline from OTL, Russian involvement in the South Seas is a long shot because setting out from Pacific ports would require some combination of much earlier than OTL arrival there, or much heavier investment in overland transport capability to there along with very heavy investment in shipbuilding and so on there. Important as Siberia is to Russia, it never was so damn attractively profitable that we can plausibly imagine this stuff to be likely. (Also, put too much independent capability on the Siberian Pacific coast, if it can be maintained there at all, and we face the possibility of a secessionist splinter Siberia--one that to be sure may well speak Russian and serve as the base for your Russo-Australia. But I don't see on what economic basis such a splinter state would stand). In LttW, Thande had very early seafaring adventurers sail all the way around Africa and Asia to seize opportunistic niches in Japan; once taken and once the influence was spread over the whole archipelago, it was relatively easy to link up to Russian Maritime Siberia even in its embryonic stage--and the development of coastal Siberia is sped up by the economic links with the Japanese protectorate. This is a very long shot and would not be plausible in just any timeline!
The question remains how plausible is it that Russians sailing out of St Petersburg or other Baltic ports (or Murmansk, Archangel, or other White Sea ports) could go all the way around the world and discover, claim, settle and hold Australia from there, with the Siberian link only coming later as a useful alternate logistical path? Well, the British did it OTL, from not much closer, but the early 19th century they had unassailable command of the seas and lots of established ports along the way. We can imagine the Russians getting an earlier start, but they always have the consideration that they have to sail past northwestern European chokepoints including the British Isles themselves as well as through Danish waters, or operate from the even more distant and ice-choked White Sea and go past Norway--and still have to go past Britain, or through waters full of British shipping and naval patrols. Weaken Britain's naval status and it becomes more plausible, or ally them with Britain which agrees to stand back and let them have Australia for services rendered. Still the Russians aren't the most likely candidates for a bunch of reasons. Not out of the running though.