Having a criminal conviction for any type of driving related offence, or any endorsement (e.g. penalty points) on your driving licence, is highly likely to affect and increase the cost of your car insurance policy. The amount by which the cost may increase will depend on the circumstances, including the type and nature of the conviction.
Insurers have the right to choose whether or not to offer an insurance policy to you if you have a conviction. Common instances where an insurer may decline to offer you a quote would be if you have (or have had) a driving ban or a drink driving related conviction.
If an insurer does offer a quotation, you may find that your premium will be quite high. However, the increase in price may not be permanent – sometimes when a certain amount of time has passed, your insurer may then decrease the price.