Let me rephrase. There will never be a situation where you are eligible for healthcare, but don't have a physical card so can't go to a doctor.
In your case, I don't know if Service Ontario would re-use your old health card number, or issue you a new health card number. Either way, you need to wait for 3 months until you're eligible to use it. So if the 3 months is coming up and you still don't have an updated card mailed to you, you can go to Service Ontario and they will give you a temporary health card on the spot (basically a piece of paper with health card number and validity date). You can use this until you get your actual card in the mail.