Mimosa, a member of my family was seriously injured in a road accident. they were unwilling to use a no win, no fee company. Instead they went to a local firm of solicitors who had a personal injury section with solicitors who specialised in this field of law, and getting a suitably specialist solicitor is essential.
They discussed the situation with her, had no doubts she had a robust and valid case, and told her that when she won the other driver's insurance company would pay their charges as part of the settlement. This did indeed happen.
However, as annodomini suggested, start with the CAB. Their address will be available online and in the telephone library or local library.