When you sign up to OVO there should be little or no downtime. Here's a summary of how long things take, taking into account your situation.
If you've asked for a new phone number when you signed up to OVO
If you have signed up to OVO and asked for a new phone number then your service will be good to go within 1 hour once you have successfully activated your OVO SIM card. Assuming that you're using your SIM card in a 4G compatible device that isn't locked to any other telco... and that you've followed all the device set up instructions.
If you're bringing your phone number across to OVO
If you're switching from another telco and you're bringing your phone number across to OVO activating your SIM card can take a little longer. In fact, if you're switching from Optus or a reseller of Optus (like Amaysim) it usually takes 1 working day for them to release your phone number to OVO. If you're switching from any other telco (e.g. Telstra), your SIM should become active within 15 minutes to 4 hours.
While that's happening, you won't be offline. You can continue to use your existing SIM card until it's disconnected from your existing telco - when that happens you'll probably see a 'SIM registration failure' error message appear on the screen of your phone. Then it's time to replace your old SIM with your new OVO SIM card. When you've carefully pushed out the right-sized SIM card (nano or standard) and put it into your phone you should be ready to go within 15 minutes.
To ensure this goes smoothly, it's really important that you have given OVO the exact same Date of Birth and / or Account number that is held by your current telco. If there is a discrepancy it will slow down this whole process. You're right if you're thinking 'that seems crazy' but it's surprisingly common how many customers we've spoken to that have had a phone number recorded in a partner’s name and details, or given an incorrect date of birth to their old telco....so please make sure your account number and date of birth match!
One last thing, there's no need to disconnect your service with your current telco - once you're connected to OVO your previous telco will automatically arrange to finalise your account with you.