You might have a chance. Someone pulling in front of you and braking hard is about the only acceptable excuse for hitting someone up the jacksy. Explain to your insurance company and let them do the rest.
Just checked the details of the other vehicle, does this make a difference?
Assuming he MOT'd it on the 16/17th the system may have not been updated yet it only expired on the 15th and its the 21st now so. See if it updates in a week