Your information will be removed from the whois database when the domain is in the pending-delete or is deleted from the registry.
When a domain name reaches its expiration date, it usually gets into redemption period for about 30 days. After this 30 days, it will get into the pending-delete state in the registry for about five days. So, for this 30+5 days, your information will still be shown in the whois database. When this time is up, the registry will delete this domain and it will then be available to anyone who wants to register the domain at any registrar.
While that domain name's still expired and not yet deleted, read your registrar's fine prints. If they say they'll keep your name on for as long as it's expired and not deleted, then there's nothing you can do short of threatening them with suit.