Although your post makes a specific reference to Australia, you also seem to be asking about general travel, so I'll start there.
Assuming that you hold a UK passport, you have the right to visit, live in and work in any of the 27 member states. Criminal convictions don't change that right.
When travelling as a tourist, some countries (outside of the EU) require UK citizens to have a visa (although many don't). With the vast majority of such countries, getting a tourist visa is just a matter of filling in your name and basic details (such as your address and passport number) on a form, and paying the relevant fee. Hardly any countries ask about criminal convictions. Those that do don't require applicants to obtain any form of 'certificate of good conduct' and they can't access UK criminal records, so most people simply 'forget' about any convictions. (Theoretically you wouldn't be allowed into Canada, even as a tourist, until you'd applied for - and been granted - 'rehabilitation'. But they don't even ask you about convictions, so you'd have no problems there. You theoretically also need to apply for a visa, even as a tourist, to enter the USA, but most people don't bother. The USA are tightening up their entry rules but they still won't have direct access to UK criminal records).
Similarly, you theoretically need to apply for a visa to enter Australia as a tourist but, if you get an 'Electronic Travel Authority' online, nobody will check up on your record.
However, Australia is unlikely to grant you a residence permit (or a temporary work permit) until your conviction is spent. Under Australian law, your conviction will not be spent until 10 years after the date of conviction (unless you were still a juvenile at the time, in which case the period is 5 years). Even then, you'd still have to declare it: