See this list for the accepted "Proof of residency in Ontario" you are allowed to use:
https://www.ontario.ca/page/documents-needed-get-health-card
As mentioned above, day 1 of your waiting period will be the date you arrive in Ontario. This can be proven with COPR if you're doing first PR landing, or airplane ticket if you're a returning PR. So there is zero rush to apply for OHIP urgently, you can take your time. Even if you apply right after the 3 months wait, you'll get an OHIP number immediately on the spot that you can use. You must apply for OHIP in person at a Service Ontario office.
You should be aware of the risk you're taking by landing at more than 5 months pregnant. In case of an early delivery or any other complications later in pregnancy, you will need to pay for all doctor/hospital costs out of pocket. You should at least ensure your wife has travel emergency medical insurance, which won't cover any regular pregnancy or delivery related costs but may help if any emergencies happen during pregnancy.
Also make sure you are intending to stay in Ontario for at least 6 months with no plans to leave during this time for more than 30 days.