A lot depends on how the UK leaves the EU (and *if*, as this is still not certain). Parliament seems to be strongly against "no deal" though that remains the legal fallback if no deal is agreed before the new leaving date of Halloween 2019. In such a case it seems likely UK drivers would need IDP and Green Cards to drive in the EU. The UK would probably continue a visa regime similar to what the EU has for the time being, it's unlikely to change in the short term simply because there isn't enough time or resources to plan anything else. The name on the visa would be UK instead of EU though. Any subsequent divergence is hard to predict but would be laid out on government websites nearer the time.

The most likely scenario is leaving with some sort of deal, and recent opinion polling suggests that the closer it leaves the country to the EU the more people in the UK would find it an acceptable compromise. Some hardline Europhobes may disagree, but I'm just reporting what the polls say the country thinks. Then it depends what the terms of the deal are, and in all probability it will be structured so as to provide maximum ease of movement between UK and EU.

However it goes I suspect people entering the UK from outside the EU won't be greatly affected. More likely, if there is no deal is that the UK/EU borders will be the source of massive queues as every vehicle and traveller is inspected.