I started the application process for City Carrier Assistant on 1/20/2016 and just received a conditional offer, so just about 6 weeks total. Still need to get the final, official offer and details on orientation.
The whole process was pretty much as everyone has already described: application, online assessment, invitation to take test, drug screen, driving history, background check, group interview. I think some areas do drug screen/background check before the actual interview, others do it after. There were only 5 of us at the group interview, I'm not sure if there were any other sessions...I think not. The interview itself was pretty easy, low key, no stress. The time really is aimed at supplying you with as much information about the job so that you understand exactly what you will be doing and answering all your questions. Everyone was great, helpful and friendly.