I take amitryptilene. I have never been drowsy from it, and have never needed to stop driving as a result. The advice is IF it causes you to be drowsy, do not drive (implied to be "don't drive when drowsy"). Honestly, medication isn't the issue - nobody should be driving (or operating machinery) whilst drowsy from any cause at all. But if someone is that drowsy then they may also not be fit for work.
In the end, all the too-ing and fro-ing about who is HR, who is OH, and what anyones elses workplace does is irrelevant. The OP's employer has offered an alternative to waiting for weeks for their OH and the OP's GP to deal with it - how long that takes is a piece of string only applicable to this one workplace. The OP is entitled to refuse the employers request. The OP can limit what the GP is allowed to say; and / or they can read the report before it goes to the employer. Or ther OP can wait and see when or if the medication has this effect and go off sick on those days. Or the OP can get on a bus or a train because driving to work is not the only option. But getting to work is the employees responsibility. Those are the employees options.
The employers options are to let them work from home, or not allow it and insist that any time the OP is unfit to attend work then they will be classed as sick. They do not have to do anything else. So if I wanted te employer to paly ball then I would be looking to find a compromise that they are happy with - or suck up the consequences of not agreeing.
This is not about what is "fair" (a much overused word - life isn't fair). It is about who holds the cards. In this case it is the employer. A reasonable employer might compromise - so they might, for example, say that if the employees doctor can confirm the actual potential for being unable to drive etc sooner rather than later, then they could look at making it work. Or you can dig your heels in, say you'll agree to an OH appointment and be off sick - or catch the bus - until that report is done.