I have just found out whilst waiting on hold the reason for the delay is a promo that AAMI has for new policies that give them 20% off. I was told it ends in about a week's time.
Not sure of specific details, but I think you just ring up and tell them that you want the 'Winter Warmers 20% off special'
This is available to existing customers that are taking out a new policy too; I did this a week ago.