I'm a Sydney boy living in Switzerland. I've flown Geneva-Sydney twice with my elderly only-slightly-mobile Swiss mother-in-law. The first time, she was in her mid-eighties and we went with her, my 3-year-old daughter and 1-year-old son, and my wife who was 6 months pregnant with #3. The second time, Grandma was in her late eighties and we had 3 kids under 6. It took me years to recover from these expeditions, but I am glad to have the chance to share what I have learned, in the hope it will be of some help to the OP.
1. As others have noted, it is possible, but Granny will need to be accompanied by an able-bodied person, preferably one with a cool head and experience of long-haul air travel.
2. She (and her carer) must go business class (at least) on a decent airline. Do not attempt to do this in economy, or even premium economy. "Decent airline" in this case means not just one with shiny planes and fancy seats, but one with frequent connections (so if you miss one, you don't have to wait 3 days or be re-routed via Mumbai and Denpassar) and a level of customer service that will allow you to speak to a knowledgeable human employee by phone in advance of travel to make sure everything is arranged and in order. I would suggest Singapore Airlines or Emirates. Get a single ticket on a single airline and avoid any code-shares. None of this will be cheap, but if your date of travel is flexible (which presumably it is in this case) there are sometimes promotional 2-for-1 "companion fares" in business class.
3. There is a question whether it is better to have fewer stops with longer flights, or vice versa. We tried both options, and while there are pros and cons of each, I think on the whole for elderly people the shorter flights are easier to manage, even given the considerable faff factor and logistical challenges of stopovers. 18 hours to Perth is too long; even the 14-hour flights from Dubai/Abu Dhabi/Doha to Sydney are pushing it. I would suggest Singapore Airlines with a single stopover in Singapore (12 hours London-Singapore followed by 8 hours Singapore-Sydney). For shorter flights, you could try Emirates London-Dubai-Singapore-Sydney, with stopovers in Dubai and Singapore.
4. I know this doesn't apply in your case, OP, but for the love of God, do not attempt this kind of trip with both elderly people and toddlers at the same time.
Good luck!