A simple answer to your question is a person can operate as a sole trader in the UK while a temporary resident in Australia. However the tax position is complicated and it is worth being clear about your situation.

There are a number of criteria which will determine whether you are resident, ordinarily resident and/or domiciled in the UK and therefore subject to UK taxation. It will largely depend upon how many days and/or part of a day you spend in the UK. So you need to keep a detailed record of the days and part days you are in the UK each week/month and be prepared to justify (to both the Australian tax authorities and HM Revenue & Customs in the UK) that you are non-resident in the UK / resident in Australia. Other factors such as the nature of the trade with UK and Australian clients and having a UK and Australian bank accounts might also contribute to being taxed in the UK.

You state you are a sole trader and therefore if you were a UK resident you would be taxed on your worldwide income. However as a non-UK resident you would only be taxed on the income arising in the UK. Your next step might be to employ a UK accountant to assess whether you are a non-UK resident and to register your UK business with HMRC.

Dependent upon the length of your residence in Australia and the proportions of trade with UK and Australian businesses you might consider setting up an Australian company. Australia distinguishes between resident and non-resident companies for tax purposes. A company is resident in Australia if it is incorporated in Australia, or if not incorporated in Australia, it carries on its business in Australia and either exercises central management and control there or has its voting power controlled by shareholders who are residents of Australia. Resident companies are taxed on worldwide income. A non-resident company (one that does not meet any of the above tests) generally pays taxes only on income derived from Australian sources.

Also see: Setting up in Australia