You can leave the investment behind in the UK, but the Australian tax system won't recognise the tax free status of the income or capital gains that pertains in the UK.
You might also have to consider Australia's Foreign Investment Fund Rules, which can tax the growth in the value of funds located outside Australia from one year to the next, even if you don't sell the investment.
If you would like some additional information please contact me.
Best regards.
Alan Collett
alan-at-gomatilda-dot-com
Registered Migration Agent Number 0102534
and a Fellow of the Institute of Chartered Accountants in England and Wales
http://www.gomatilda.com and
http://www.collettandco.co.ukOffices in Southampton, England; Perth, Australia; and Melbourne, Australia