We decided to book an appointment for the next day (today) and booked the flights! In the end it was £779 each, which isn't so bad for three flights. We'll leave from Heathrow at 20.40 on the 27th of August, and we will arrive on the 28th in Dubai at 6.30. We will spend 2 nights in Dubai and on the 30th we'll leave at 9.35 and arrive in Singapore at 20.55. We'll be there until the evening of the 3rd of September and we'll arrive in Melbourne at 7.40 on the 4th of September.
It's a lot of flying, but it'll be worth it. We'll be able to see all of these cool places and it means we don't have to be on a plane non-stop for 22 hours!