UUPPSSS change of situation as of today:The TRP application process is outlined in the link below. Since you are visa exempt / required an ETA, you'll need to speak with the visa office for your region to confirm how to apply.
https://www.canada.ca/en/immigration-refugees-citizenship/services/immigrate-canada/inadmissibility/temporary-resident-permits.html
How long it takes to process is very difficult to say since processing times vary widely. It's possible you may hear back after a few weeks - however several months is also entirely possible and quite common.
My wife went to her MP office to demend speedy aproval of TRP,,, after one hour on the phone the MP informed my wife that acording to imigration Canada i am still Permanent Resident,acording to CBSA i am not because deportation order has been issued but never enforced,so i guess i have to go back in order for them to deport me and then apply for TRP,and apperantly Imigration Canada and CBSA don't share info.In any case i am not allowed to go back so i can get legaly deported and i am not alowed to apply for TRP go figure......