The process of filing an asbestosis lawsuit will vary from country to country.
In the UK asbestosis is considered an industrial disease. As such prior to starting a claim for compensation at court the process follows the Industrial Disease Pre Action Protocol and when issued in court follows the Civil Procedure Rules.
I have listed some helpful articles below describing how to claim compensation for various industrial diseases including asbestosis as well as the amounts of compensation you can expect to recover for an asbestosis compensation claim.