My wife was driving my daughters' car recently and was hit in the rear by another car causing major damage to the rear offside. She was also pushed into the car in front, creating some minor damage/scuffing. The driver of the car who hit her immediately admitted full liability to the police. Our insurers Admiral have said the car is a write off but are being very slow to make an offer to us. Meanwhile LV, the insurers of the third party, have been to take pictures and have made a good opening offer for the car saying its a Cat C right off. Why would LV try to settle directly and what is the downside of accepting any such offer?

How is this no fault claim likely to affect future insurance premiums for my daughter (as the policy holder) and my wife as a named driver involved in the incident?