So far we know that there's two stages: eligibility and admissibility.
With eligibility it seems like we only need to be able we meet the program requirements, so I'd say just work/school documents, language tests and the basic PR documents. This is the equivalent of the R10 check for EE.
If we pass eligibility, it looks like we'll be moved into admissibility. This is where we might get ADR request such as medicals and PCC and basically prove we're not a threat to the country.
IMO, if you have both eligibility and admissibility documents upfront your application has good chances of being processed fast, it's been hinted by several sources (leaked briefing, chatty lawyer, CICnews...) that this program is a once in a lifetime thing to meet 2021 immigration quota, this leads me to believe they're expecting to process those applications pretty quickly.
So if you have everything upfront, have a simple profile and apply on May 6th, I see pretty decent chances of getting a COPR by September-November.
This is just speculation though, we'll have to wait and see