Your situation brings to light a disconnect between government agencies in Canada. Assuming rules in ON are similar to BC, you technically have to exchange your DL after 30 days of moving into the province. My Inland PR application took more than a year, and IRCC suggested I not exchange my DL until the application was approved, so I maintained my US DL the entire processing time. Once my application was approved I exchanged my DL without issue.
So, to answer your question, assuming her current DL will remain valid the entire processing time, I'd suggest you wait until the application is approved. If your wife gets pulled over in the meantime she could simply explain the reason she's waiting to exchange her DL.