I doubt it.
Unless your drive happens to be a route that the public have access and use of on a regular basis. Even then I suspect a police officer would have a hard time getting it into court, but the moment your wheels touch say the public footpath to cross it and reach the road, or the road itself you'd definitely have committed an offence.
IIRC the distinction is that you don't invite the public onto your driveway and give them unfettered access to it (the public have no expectation to be able to use your drive), the only people who'd be expected to be on your drive would be invited or deliveries etc.
Unlike say a shop's car park, or the drive through at McDonalds.
There is a lot of case law about this, ranging from things like private roads to communal parking/housing estates, to car parks, to farmers fields with a public event going on in them.
A drive through would be either classed as a "private road" that the public have access to, or part of the retailers car park, both of which are places where the Road Traffic Act and related laws have been deemed to apply.