I hope you don't mind me throwing a non-Disney/Universal curve ball, but for young kids I can recommend going to Seoul, South Korea. This is a good (cheaper) trip suggestion for everyone that isn't AlexB as well.
First, the country:
It's much cheaper than Japan, Singapore, or HK.
Seoul is very English Friendly (subways and buses all have English announcements)
Korean Air have direct flights from Brisbane, Asiana have direct flights from Sydney.
Cathay have really cheap flights if you don't mind stopping in HK for a few hours.
There are 3 theme parks in Seoul.
Lotte World - http://adventure.lotteworld.com/eng/main/index.do
The world's largest indoor amusement park.
Dozens of rides, (mostly moderate thrills), but there is a lot for small kids as well as adults, and some good family attractions
They do 2 Parades each day (lunch time and evening). The evening parade is pretty incredible (Disney-level good).
It is literally in the middle of Seoul (Jamsil) and attached to Lotte World Mall, so there is plenty of cheap food and accommodation nearby.
Discounts for Foreigners (a bit of research to find the right deal and you can get in for as cheap as $15-20)
Has an Ice skating rink in the middle of it (with free skate hire for foreigners).
Everland - http://www.everland.com/mobile/english/
Samsung's Theme park.
About 1-2 hours just outside of Seoul
Is a resort, so you can stay on-premises if you want or stay closer to Seoul and get the bus.
Has a water park called Caribbean Bay
Is also a zoo with safari-style attractions
The park is massive with great theming and really good attractions (including T-Express an Intamin Woody)
Very family friendly
Beautiful scenery in the Spring/Autumn if you can go then.
Tax Free shopping for tourists
Seoul Land - http://eng.seoulland.co.kr/eng/index.asp
A much older park (built for the Seoul Olympics)
Very family friendly (really cheap food, great rides for kids and families)
Only a short 15-20 min train ride from Central Seoul
In the middle of Seoul Grand Park (which itself is really nice)
Really cheap entry - and some websites give discounts if you enter after 12.
Not as English friendly as the other two parks (really made for locals), but still easy enough to navigate.
To be honest, you could take the kids to Korea when they are any age and you'd still be able to get a lot of value out of it. You could also do all three of these parks in Seoul and still have a cheaper trip than going to Japan.
It's not Disney or Universal, but it's a great holiday option that a lot of people don't think about.