I'm assuming your partner's British passport is because he has UK citizenship (in addition to Australian).
As an Australian citizen you don't need a visa to enter most European countries for stays up to 90 days per rolling 180 day period, but you will need a residence permit if you stay in, say, the Schengen area beyond that limit. (Which sounds likely to be the case unless you're planning your itinerary specifically such that you're outside Schengen for enough time to satisfy the 90/180 rule -- in which case you'll still need to deal with the rules of those other countries).
An UK youth mobility visa will not in itself count as a residence permit for other European countries.
The good news is that as long as you're accompanying your partner who is exercising his citizen's right of free movement within the EU/EEA, you basically cannot be denied a residence permit. You still need to apply for one, and there will be some processing time, but in most cases it will be fast-tracked, and most of the regulatory barriers to other prospective immigrants will not apply.
In principle the rules apply to cohabiting partners, but it will probably make for smoother trips through the bureaucracy if you get married before you travel. (Nobody says you need an actual big wedding, though).
[With the length of trip you're sketching here it is plausible that you will need to deal with tax authorities in the countries you're staying in, even if no money change hands -- but that is outside the ambit of Travel.SE]