Generally it seems to be (at least for the police check) fairly soon after linking the application on MyCIC. If you need a police check that takes a while, you're going to want to initiate the process well prior to this point to account for that (I'm thinking specifically of non-US citizens or PRs needing an FBI check from the US, there may be other examples but I'm unfamiliar with the time it takes to obtain police checks from other countries).
As for medicals, just from casual observation it seems to be coming after sponsor approval and/or shortly after the application gets transferred to a VO. In other words, so stupidly soon there's no reason they shouldn't still allow them as an upfront option anymore.