Police checks from Canada aren't required at time of application, and when you get one, you don't always need fingerprints if your name isn't common. Try to apply through your regional police for the criminal record check (it says 2-3 weeks processing time), but if they do ask for fingerprints then yeah I don't know how long it'll take.
"A criminal record check will determine if you have been charged or convicted of a crime. If a name-based criminal record check does not provide a definite way of confirming your identity, you may be asked to provide fingerprints. The use of fingerprints is the most accurate way to confirm a person's identity. They ensure that individuals cannot evade their criminal past, and protect individuals from being falsely associated with a criminal record that is not theirs."
Source:
https://www.rcmp-grc.gc.ca/en/steps-getting-a-certified-criminal-record-check
Also, for the application itself..
"Police certificate and IRCC requirements
You don’t need to provide a police certificate when you apply for permanent residence in Canada. We’ll tell you if you need a Canadian criminal record check while your application is being processed."
Source:
https://www.canada.ca/en/immigration-refugees-citizenship/services/application/medical-police/police-certificates/how/canada.html
So this is IF they ask for it later.