Can you get married next week? Or even the week after? Well... the short answer to this question is "Probably not."
Couples need to give their celebrant their Notice of Intended Marriage (NOIM) no earlier than 18 months and no later than one month before their wedding date. A prescribed authority may authorise a marriage with less than a month's notice; however, there are only five reasons they may do this:
employment-related or other travel commitments
wedding or celebration arrangements, or religious considerations
legal proceedings, or
an error in giving notice.
If you need to apply for a shortening of time, you will need to have your NOIM completed and witnessed and given to your celebrant. Your celebrant will then refer you to a prescribed authority. You would need to make an appointment with the authority and take your NOIM to them. You may also need to take proof of your reasons for applying for the shortness of time, for example:
letter from a medical practitioner
receipts for payments made
The prescribed authority has no discretion to approve shortening of time outside the circumstances listed above, and there is no guarantee that the shortening of time will be automatic.
A list of prescribed authorities can be found on the Attorney General's website.