Experienced injury attorneys have ways of dealing with this challenge, but it surely is better not to have this gap in treatment in the first place.
Getting x-rays of your neck and back even after a relatively minor accident can only benefit you. If the findings are negative, this will give you a peace of mind that you don't have any serious dislocation in spine. If there are objective findings of back or neck injury, this will allow your doctor to prescribe a more effective treatment plan and provide you with options for pain relief, as well as possibly prevent further deterioration of your condition.
And of course, getting treatment promptly after the accident will take away the above "gap in treatment" argument that the insurance companies love to use both in settlement negotiations and in court.