You will need to discuss the damage done at the accident scene and possible motor damage with the claims adjuster from the at-fault's insurance company. Did the police come to the scene and determine your car was able to drive away? If so, this could strengthen your case since others did not recognize the radiator issue that was realized on the way to the repair shop.

Before speaking to the claims adjuster you might also want to get documentation from the mechanic stating that the damage done was a result of the accident. Negotiating An Insurance Claim Settlement is an article you might find of interest.