I was a little confused - Do you mean that you e-submitted your AADSAS application well before completing your DAT, or that you just sent them off the entire package not too long ago ? I called up prometric a while back, and they told me it typically takes about a month after the date of your examination for the report to be sent out to various destinations. So maybe a little after a month from your test date for the review to begin ... ? Just my guess.
If you're looking to start the process so that you aren't caught too late, just fax off your score sheet
to schools that would accept it. Some schools made it clear that they still wanted the official score report, but were fine with looking at an unofficial score sheet for initial review. Perhaps some of the schools you applied to would be amenable to it as well.