You are very close to asking for financial advice, which no-one on this forum is allowed to give. If, however, you’re asking a general question about how long it takes for a CC cancellation to be reflected on one’s credit rating, my experience is that it takes between one and three months. I suppose it depends on how long it takes for the bank or CC provider to get around to telling the credit rating company that the card has been cancelled.
Also speaking generally, someone with only one credit card, easily-manageable debt and a permanent job shouldn’t have all that much trouble getting approved for a new credit card (supplementary cards don’t count when banks are determining whether to approve a new CC application).
If you’re asking how long you need to wait before you can take advantage of any bonus point offers for new cards, off the top of my head with both ANZ and BoM you need to wait 12 months before applying for a new card.
If you apply for the AmEx, make sure you get the primary cardholder to refer you, so they’ll get bonus points for the referral.
Hope this helps!