Here is a reply I got from Guy Verhofstadt on this issue:
Thank you for writing to me again and for sharing your concerns.
I fully understand your uncertainty but hopefully I can alleviate your concerns, at least in relation to the recognition of driving licences. The reports you read in the media are based on a presentation from the European Commission which takes the basic starting point that the UK becomes a third country after the Brexit date. The consequence is that the UK exits the internal market for road transport and that the current EU-law based rights, obligations and benefits cease to apply. Among those benefits is the mutual recognition of driving licenses.
However, as also pointed out by the European Commission in the presentation, the UK ratified the 1949 Geneva Convention on Road Traffic, which implies that UK driving licenses will still be recognized in countries that are party to this agreement. This includes almost all EU Member States, except for Germany, Croatia, Estonia, Latvia and Lithuania. In addition, when the UK ratifies and transposes the Vienna Convention on Road Traffic into national law in the future, those missing Member States would also be covered.
As you can see, in this particular case there are fall back positions secured by international law that can protect the recognition of UK driving licenses in the EU Member States. However, you are absolutely right in saying that citizens should not be negatively affected by this and the European Parliament will fight for them to make sure they do not become the victims of this process.