M6 crash 3rd party disputing liability

Hi, just looking for some advice.

Back in April I was travelling in lane 3 of the m6 southbound overtaking a lorry. The traffic I'm front stopped suddenly and I braked really hard and managed to slow to almost a stop ( the car infront pulled to the right of lane 3 practically in the Centre barrier) whereas I stayed to the left of lane 3 and managed to not hit them. The car behind me just clipped the back right of my car with his front left bumper. Not much damage and noone was hurt but we exchanged details and I informed my insurance immediately.

My insurance contacted the third party's insurance and they've neither admitted or denied liability until now. I got a letter from the third parties insurance blaming me. Stating I suddenly pulled into lane 3 and caused the incident. This is not true. My question is what do I do now? How can they lie in this way? I don't want an at fault claim against me and unfortunately there's no witnesses to corroborate either sides versions or events. Will the third party get away with it?

One very unhappy camper here. Any advice would be welcome. I've contacted the highways agency to see if I can get hold of traffic camera recordings but I think that's a long shot 😢

Thanks