Julie,
A wrong information supplied on the car insurance application form can lead to your policy becoming void. If they ask for any convictions for a named driver to be added to the insurance policy, then you should ask the person concerned for the correct information. The policyholder is responsible for the policy and is therefore liable for accurate data being provided for the purpose of insurance cover. Whether you didn't know about the convictions or have deliberately omitted these details, the insurance company will not accept any responsibility for this and you will be liable for everything.
When you take out an insurance policy, they send out a policy document that you need to check for errors and inform them accordingly. If your circumstances change, for example if you get points on your licence, you should inform your insurer as well. Keeping the policy up-to-date and accurate is your responsibility and the insurer will use this against you. So I doubt you stand a change in proving them wrong or getting away with not declaring the conviction of driving without licence for the other driver.
