Lovely that your Gran wants to go to see her sons and family - However it seems she will need to be accompanied by a relative, or carer on the flight, plus her Doc has to say she is OK to fly or go by sea.
QF say You must travel with a carer if you're unable to self-toilet.
The non-stop QF LON to PER is possibly one easier hop....They have a Disabled/accessible loo on their 787 in the centre of the cabin.
PS I am in Dorset too -
My first thoughts were to take a Cunard ship to down-under but the Queen Elizabeth sailed last Sunday to Fremantle which was a quick passage - 28 days
The Queen Mary sails on 10 JAN 2020 to Oz via the far east to Darwin Brisbane then SYD - but a long voyage 48-56 days
P&O may too have a voyage from SOU.