The German federal government has announced its plans for UK nationals in Germany after 29 March 2019, in the event of a no-deal Brexit.
National residence and/or work permits will be required for UK nationals who wish to continue to reside and/or work in Germany after a no-deal Brexit, as they would be considered third-country nationals.
However, UK nationals will not need to leave Germany by this date, but will have a three-month grace period in which to submit an application for a work and/or residence permit with the relevant local immigration office. If they have not already done so, they will also need to register with the registration authority at their place of residence. They will then be able to continue to live and work in Germany (along with their family members) until their application has been decided.
Many details of the planned transitional arrangements remain unclear, for example:
Source: Peregrine Immigration News