Whenever you suffer loss or injury through the negligence of another person, you're entitled to claim damages from them. That applies irrespective of whether it's a road accident and irrespective of whether or not the person has insurance cover.
Your claim is 'officiallly' against the driver but, in practice, it will be handled by his insurance company. Contact the driver's insurance company, as soon as possible, notifying them of your intention to make a claim against his insurance. Give details of the accident and state that you will be seeking quotations for the repair of your bike and that you will be forwarding them shortly. Also, invite the company to make an offer for compensation in respect of the shock and injury to yourself. (Send a copy of your letter to the driver).
Get a specialist cycle shop to give you a written quote for repairing your bike. (If possible, get several quotes. insurance companies usually ask for 3 but they might be prepared to accept fewer if you live in an area where there are only one or two firms who could do the job). Forward these quotes to the insurance company. (Once again, send a copy to the driver).
Hopefully, that should be all that you need to do. You definitely DON'T need to use the services of one of the many firms of 'compensation lawyers' who advertise in the media UNLESS you fail to get a satisfactory response from the insurance company.
Chris