Hard Brexit and travel for UK citizens into Schengen countries in 2019

You will certainly¹ not need a visa. The UK has declared it will not require visas for any EU nationals for short stays, and the EU has declared the same for the UK (see European Commission press release and Brexit preparedness document). However, from 2021 you will most likely need to apply for ETIAS (an electronic authorisation similar to US ESTA). This is likely true even if there is a Brexit agreement and possibly even if there is no Brexit (without Brexit, EU law should mean no ETIAS shall be needed).

The UK Government has issued a series of notices on the impact of a no-deal Brexit. One such notice issues specific guidance for Travelling to the EU with a UK passport if there’s no Brexit deal (checked 13 December 2018). This confirms the guidance quoted in legoscias answer, but does not address a system known as ETIAS.

Once it enters into force (expected 2021), UK citizens will most likely need to apply for ETIAS, a Schengen electronic system similar to the more famous US ESTA system. See ETIAS VISA: how will it affect UK citizens:

Firstly, anytime a British citizen visiting Europe with a valid passport wishes to travel to continental Europe, prior to leaving home, they will have to apply via an online platform for an ETIAS visa-waiver.

After the UK leaves the EU, britons will be exempt from obtaining a visa, but will not be exempt from the ETIAS visa waiver.

This travel authorisation, once accepted, will be binding for 3 years for multiple visits with no limit and will become a mandatory requirement for all travelers from elegible ETIAS visa waiver countries from 2021.

UK citizens and all eligible travellers who wish to visit the Schengen Zone will need to answer a series of health and security questions when completing the ETIAS online form. It’s extremely important to answer honestly, as all the data provided in the ETIAS application will be checked against a series of databases including Europol or Interpol.

This statement from the European Commission (which the same website cites) confirms that according to stated policy, travel in both directions will remain visa free:

It would mean that UK citizens would not need a visa when travelling to the Schengen area for short stays of up to 90 days in any 180-day period. In the scenario where the UK leaves the EU without a deal, this would apply as of 30 March 2019.

(…)

This proposal is entirely conditional upon the UK also granting reciprocal and non-discriminatory visa-free travel for all EU Member States, in line with the principle of visa reciprocity. The UK government has declared its intention not to require a visa from citizens of the EU27 Member States for shorts stays for the purposes of tourism and business.

So, unless either party changes its mind and breaks with stated policy intentions, you will not need a visa. However, from 2021 you will need ETIAS.

Edit: The day after I posted this answer, The Guardian posted this "news" article on Brits needing to pay the ETIAS fee for Schengen travel after Brexit, which is old news because this was already known.

¹Of course, in theory, any government can change any policy at any time, reasonably or not, legally or not. This answer is accurate at the time of writing. Any country might suddenly close their borders after you book but before you travel. That is a risk that always exists for future travel.

The UK passport office has published a document titled Passport rules for travel to Europe after Brexit. It suggests two things:

After 29 March 2019:

1. You should have at least 6 months left on your passport from your date of arrival. This applies to adult and child passports.

2. If you renewed a 10 year adult passport before it expired, extra months may have been added to your new passport’s expiry date, making it valid for more than 10 years. Any extra months on your passport over 10 years may not count towards the 6 months that should be remaining for travel to most countries in Europe.

Those points apply to Schengen countries only. For non-Schengen countries:

The new rules do not apply when travelling to Ireland.

Bulgaria, Croatia, Cyprus and Romania are not in the Schengen area. You should check the entry requirements for these countries.

The word "visa" does not appear in that document, so it appears that the UK government believes that the EU will let UK citizens visit the Schengen area for up to 90 days visa-free.

Edited to include link and quote of EC press release on the topic.

The problem with visa policies is that they are very often governed by reciprocity: if you don’t let my citizens come to your country visa-free, I won’t let yours come to mine visa-free (of course this usually does not apply to countries which depend on tourists, for instance).

So, even though it is widely expected that the EU will add the UK to the visa-exempt list, they will not be very inclined to do so if the UK does not grant visa-free travel to all EU citizens.

Actually, they did say exactly that here:

European Commission proposes visa-free travel to the EU for UK nationals in a no deal scenario – if the UK also grants reciprocal visa-free travel to all EU citizens

(…)This proposal is entirely conditional upon the UK also granting reciprocal and non-discriminatory visa-free travel for all EU Member States, in line with the principle of visa reciprocity.(…)

(thanks @gerrit for the link)

And that’s where the problem may arise.
As soon as those Brexiter hardliners manage to have the government say "sorry, but citizens of (pick any of the poorest EU countries with a history of economic migration to the UK) can no longer come to the UK visa-free", the EU will probably reply "sorry, no UK citizen can come to the EU visa-free" (note that this does not apply for existing relationships, like EU-US, but as a matter of principle, I’m pretty sure they will have a hard time accepting it in this case). Then it’s a matter of deciding whether principles or the economic impact matter most, and who will chicken out first.

The same release says:

The UK government has declared its intention not to require a visa from citizens of the EU27 Member States for shorts stays for the purposes of tourism and business.

Now, between "declaring its intention" and actually implementing this into regulations, we have a few obstacles. We can hope reason will prevail and this will actually happen, but at the moment it’s not a done deal.

So you may need a visa. Or you may not. Nobody knows. And in the meantime, you can’t get a visa. Sorry to say so, but pick a destination outside of the EU if you want to be safe.

The only thing you can know for sure right now is that you currently can’t get a Schengen visa. There is no point in applying for one to be on the safe side. Schengen visas are only issued to nationals, who actually require a visa to enter. Visas are not issued to EEA citizens and citizens of visa exempt countries.

On the list of visa exempt countries, you find since May 2014 the following category:

British citizens who are not nationals of the United Kingdom of Great
Britain and Northern Ireland for the purposes of union law:

• British nationals (Overseas)
• British overseas territories citizens (BOTC)
• British overseas citizens (BOC)
• British protected persons (BPP)
• British subjects (BS)

All British nationals who are currently not EU citizens are already now entitled to enter the Schengen area without a visa. If we should get a new category of non-EU British nationals on March 30th next year, I don’t see why they should not already be covered by this exemption.

You’ll be able to visit without a visa for up to 90 days within a 180-day period.

Can’t prove it in the strictest sense, but although the EU may threaten with something else as part of the political game (the US has done this to the EU as well) there’s simply no way it’s actually going to happen when other low-risk nationalities are visa-exempt. It would be the highest degree of political idiocy.

Strictly speaking, this is in the hands of the EU, which maintains two lists, Annex I (countries whose nationals requiring a visa for a short stay, labelled A in the link above) and Annex II (countries whose nationals do not require a visa, labelled B). As a member state, the UK is not on either list, and it falls between two stools in terms of automatically being added to one or the other, so will require EU action fairly soon, otherwise it will be legally impossible to visit under either set of terms.

It’s very widely expected that the UK will be added to Annex II regardless of whether a formal exit agreement is reached, and you should expect a decision on that well before March 29th 2019, but as to when it will happen, that’s rather a victim of circumstance.

I don’t think anyone knows how yet how this will all work out. One link suggests that there will be free-movement transition period until 2021
https://www.independent.co.uk/news/uk/politics/brexit-latest-visas-travel-free-movement-eu-commission-france-schengen-a8595991.html

On the other hand, some British airlines are nervous and Ryan Air has already warned of significant impact on stock price
https://www.independent.co.uk/travel/news-and-advice/brexit-flights-29-march-2019-flights-passport-customs-uk-eu-roaming-ehic-a8531866.html

I doubt there is anything pro-active you can do at the moment. Even if you apply for a Visa, the issuer wouldn’t know what to do with your application since the rules aren’t clear yet.

Other reading

https://www.schengenvisainfo.com/no-visa-free-traveling-brits-if-they-exit-the-eu-without-an-agreement/
https://www.schengenvisainfo.com/likelihood-of-post-brexit-uk-falling-under-etias-program-sparks-fury-among-britons/