new driver or newly-passed driver

You should say you are a new driver. Unless unlicensed driving is the common thing in your country, (I doubt that is the case), the listener will assume you first passed your driver's test before you started driving.

Thanks, Blickers.

You're right. There is no unlicensed driving where I live.

In Britain, being a new driver starts as soon as you drive under supervision. To learn to drive you need a 'provisional' licence*, which carries restrictions - you have to be accompanied by a fully licensed** driver aged at least 21 who has held a full licence for at least 3 years. You have to display white plastic plates with a big read L (for 'learner'} at the front and back of the car. You are a 'new driver' as soon as you start taking accompanied driving lessons. Once you pass your test and get a full licence you can (but don't have to) display plates with a green P (for 'probationary') to indicate to other road users that you are still a 'new driver'. Such people are referred to as 'newly qualified drivers'.

*By the way, that is how we spell 'licence' (noun) in Britain.
** And that's the verb.