If the UK leaves the EU without any deal (the so-called "no-deal" or "Hard" Brexit) and it is not placed on the Annex II list of countries of the Schengen agreement, would UK citizens begin to formally require a Schengen visa to enter the rest of the EU? If so, are there any plans to add the UK to the list of visa-free Schengen countries outside of a formal deal with the EU?
The EU is looking into its visa policy in light of the brexit vote. Whether or not this means UK citizens will need a visa post-brexit depends on the outcome of UK-EU negotiations. Politico has an article about this, some relevant quotes (I suggest reading the full article for more background):
The European Commission is making plans that could mean British citizens need a visa to enter the EU after Brexit, according to a document seen by POLITICO that was presented to MEPs.
Whether it comes into force, or an alternative that would exempt U.K. citizens from visa requirements, will depend on the outcome of the Brexit negotiations. But Selmayr told MEPs that the legislative changes, in eight different policy areas, are needed for the EU27 to avoid disruption in the aftermath of the U.K. leaving the bloc.
When there is no deal
A ‘no deal’ scenario is one where the UK leaves the EU and becomes a third country at 11pm GMT on 29 March 2019 without a Withdrawal Agreement and framework for a future relationship in place between the UK and the EU.
In that case, it would be up to the EU to require a visa or not (probably by adopting some of the plans mentioned in the Politico article). The EU writes the following about its visa policy:
The EU aims at achieving full visa reciprocity with the non-EU countries whose nationals are exempt from the visa requirement. Thus, EU citizens would not need a visa either for travelling to these non-EU countries.
With that in mind, the EU is likely to require visas if EU citizens face the same requirement when travelling to the UK. In any other scenario (e.g. it's not clear what the UK's policy will be) it's pure speculation what the EU will do. The fact that the EU is creating different plans shows that they probably haven't decided themselves.
Technically speaking, we are British citizens not UK citizens, this may seem like pedantry, but it's actually relevant because of the complex nature of British and EU citizenship. British citizens who are not UK-nationals are already covered in the EU laws (see here) and require a visa or not depending on their status. In the absence of any deal this would continue unless the EU decided to change it.
But for the rest of us, the situation is undefined. Rather extraordinarily, the rules are drafted in a way that has both a complete list of countries that require a visa (Annex I) and a list of countries that don't (Annex II) rather than defaulting to the former status if not listed. The status of countries that are on neither list (as UK-national British citizens would be in the circumstances given) is undefined in the agreement.
I am unable to clearly determine what this undetermined status would mean, but my interpretation of the community code on visas leads me to believe that it would not be possible for UK-national British citizens to either get a Schengen visa or to travel within Schengen without a visa. I think it would then only be possible to get a territorially-limited visa for a particular country within Schengen since countries still have exceptional power to grant these.