This depends entirely on the issuing institution's own process, which can range from quick to genuinely slow, especially if the institution has closed or merged. Start as early as possible and treat this as a prerequisite to travel or filing, not a parallel task.
Start the records process well ahead of any filing date
Because institutional response times aren't within the applicant's control, begin requesting replacement or official records as soon as a gap is identified, rather than after a filing date is already set. Build the rest of the TN timeline around when verified records are realistically expected, not the other way around, to avoid pressure to file with documentation still incomplete. There is a sequencing point that shortens most of these timetables.
Send the institutional request on the day the gap is identified, before the profession's requirement has been confirmed, because the two enquiries run in parallel and only one of them depends on an organisation that answers on its own schedule. While the request runs, assemble everything the applicant already holds and confirm the profession's exact requirement, so that the moment the records arrive the comparison can be made rather than begun. A start date agreed before the institution has even acknowledged the request is a date built on nothing.