The status of British nationals in EU countries after Brexit will depend on if a deal is agreed between the UK and EU. Until the UK leaves the EU, the UK remains a full member of the EU and UK nationals will keep their legal status as EU citizens.
Brexit withdrawal agreement
If a deal is approved, the Withdrawal Agreement will secure the rights of 1 million UK nationals living in the EU. The Transition Period will allow UK nationals in EU countries to live largely as they do now until the beginning of 2021. The Government of Jersey has legal clarification that the references to 'UK nationals' in the Withdrawal Agreement refers to all British nationals. If you have a Jersey-issued British passport you will be covered under the Withdrawal Agreement. For nationality purposes there is no differentiation between a British national resident in the UK, Jersey or anywhere else in the world, or where their British passport has been issued.
Currently those holding Jersey variant British passports without the 'Islander endorsement', can work and reside anywhere in Europe because they are British, as they are exercising their EU Treaty Rights. After Brexit, the 'Islander Endorsement' will no longer be put in any passport as there will be no legal basis for it.
No deal Brexit
However, in the event of a Hard Brexit, or Day One No Deal, living and working in an EU Member State will depend on the EU and its Member States, and whether they reciprocate the UK's offer in guaranteeing the rights of EU citizens to continue living as they do now.
The EU Commission has urged Member States to take a generous approach to British nationals who are already resident in their territory, a number of which have indicated they will. Any British nationals resident in the EU should contact their local authorities and be ready to cooperate with them once they confirm any action they may be required to take.
The UK Government has more information advising British nationals who are currently living in the EU.
Advice for British nationals travelling and living in Europe on gov.uk
Guidance for UK nationals living in Europe if there's no deal on gov.uk
Overseas living guides on gov.uk