Hi,
You can be tax resident in both the UK and Australia in the same tax year. As you are considered tax resident in the UK for the whole tax year, you will need to then consider if split year treatment applies to your circumstances. You can see the guidance on split year treatment and apply the rules to your situation here:
RDR3 Statutory Residence Test
If split year treatment applies, then in order for you to claim it, you will need to complete a SelfAassessment tax return, SA100, and the relevant supplementary pages, including SA109, residence, and form HS304:
Non-residents tax relief under double taxation agreements (Self Assessment helpsheet HS304)
The tax treaty in place with Australia allows the Australian tax authorities to tax your UK teachers pension:
UK/ Australia Double Taxation Convention
For future tax years, you should download and print off DT individual UK / Autralia form:
Double Taxation: UK-Australia (SI 2003 Number 3199) (Form Australia-Individual 2003)
The completed, signed and dated form, should be sent to the Australian tax office for validation. They will return it to you stamped and signed. You should then send it to HMRC at the address on the form.
Thank you.