Third Party Motor Insurance Claim

 My car has been damaged in road accident and the other party has asked me claim the damages through their insurance, the process for which as I understand requires an FIR and court case. Could you please advise how long would this process take in court for a decision to be reached and will this be an extremely cumbersome task to pursue. My insurance will not cover the full extend of my damages and I do not want the other party responsible of negligent driving - slept at the wheel. (DMC milk van) to get away with this.