I suspect you haven't written to them in the correct way. If you'd written a properly drafted letter of claim, they'd have to notify their insurance company or risk picking up the bill themselves. As the pro's have said, there are plenty of firms who will consider this without charging you up front. It may be unwise for you to attempt a letter of claim, unless its something you've done before. Particularly since personal injury claims are a bit more complex than damage to property.
I wouldn't read anything in to the change of driver.