I was contacted by someone who is struggling to nail the Cockney accent, and he keeps slipping into what sounds more like an Australian accent. This isn't uncommon at all! The two share many sound changes - the diphthongs are especially similar.
This is also an issue that I've run into a great deal with students and performers, so here are some tips on things that may help you to keep from going down under with your London accent:
One former student even seemed to slip a bit into some American Southern elements when working on this accent, while also drifting between Australian and Cockney, or, as she liked to call it, South Cockstralian. This seems on the verge of NSFW...