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/immigratio...admissibility/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.
UUPPSSS change of situation as of today:
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......