I am not a driver, but I still think you need to tell them, and it’s not just a matter of personal discretion whether you choose to or not, as their insurance may not cover you if you had an accident, and it turned out you had an undisclosed medical issue.
I do not believe you will be barred from driving the company car, or that the company will be charged more for your premiums, as either would be discriminatory.
I had a similar situation when it came to claiming critical illness insurance. My cover was provided through work, and the claims procedure was…through work.
Which, in my view, was wrong, as it meant I did not have the option of choosing to maintain privacy concerning my diagnosis - not if I wanted to collect the money, that is.
Overall, I do think it’s better that work know, especially once you have a confirmed diagnosis - which I appreciate is not the case for you. Without them knowing, you do not have the full protection of anti-discrimination legislation.
However, although I would almost certainly have told HR anyway, I did object to the fact I didn’t get a choice about it. There was an awful lot of money at stake - money I could not realistically afford to pass up on a point of principle.
Unfortunately, sometimes, you can be put in the position (however unintentionally) that there isn’t really a choice. I suspect this may be the case with you and the company car.
Somewhere, there is probably a company car policy, and an H&S policy, one or both of which sets out the responsibilities of the employee regarding disclosure of any medical issues for insurance purposes. I’d be very surprised if it wasn’t set down in writing anywhere. Whether you can get your hands on it without tipping them off why you want it is another matter…