I mean Australia is given 50/50 odds in most COG scenarios. It is geographically remote, removed from the jetstreams which means reduced fallout risk, has defense and resettlement agreement with multiple island nations which could augment the food supply and labor force in a world without international trade as we know it, and has a military and government capable of rebuilding a sizeable portion of the Pacific's infrastructure after a nuclear war.
But there is also the fact that a lot of countries, like China and Russia, have factored Australia into their first and second strike strategies and would lob a few ICBMs at major cities and installations to ensure the Australians do not inherit the Earth .... but it's an open question whether they would want to waste reaction time responding to a potential future threat when they could try (and probably fail) to intercept incoming nukes.
Either way you dice it, the bad news is most of the world won't be happy campers come morning. The good news will be they won't be unhappy campers because they'll be dead.
that's mostly all science fiction stuff
as for the fallout look up the 7/10 rule its not actually going to take 200 years for fallout to clear nor will it completely envelop the world at ground zero you'd be able to go outside within two weeks the important thing to remember is a nuke strike is only one blast of radiation its not a nuclear plant constantly leaking radiation
as for Australia being hit well it wont what i'm talking about is not an assumption that Australia is not important enough to be hit its the actual real world prediction coming from military advisers analysts etc etc Australia just wont be hit if we do it would only be one site pine gap which is far far far away from anything
and finally nukes are not going to be just lobbed indiscriminately nukes while they do a lot of damage they're not these planet destroying death balls like movies and media portray them in reality nukes are used ahead of a massive invasion the idea is to knock out a countries defences and then invade before a counter strike is possible in the case of MAD its a cunt move but if you don't live near major urban centres just stay in doors for two weeks
put it this way when the oil wells went up in iraq everyone predicted the end of the world it would trigger a winter for 200 years and spark an artificial ice age etc etc
all that happened is some soldiers got lung conditions later in life environment was not effected
the idea that fallout will gas the world comes from misunderstanding how radiation works its not fire just because you have more of it does not make it any more powerful its still going to reduce very quickly you don't need a vault tech vault to survive it either you just need to ductape any gaps in your house and wait out for 2-4 weeks