You may probably know that, as a British citizen, you share, with the nationals of few other countries, the privilege of being able to apply to the local immigration office in Germany for a residence permit after arrival in Germany and without having obtained a visa prior to travelling to Germany.
The residence in Germany must be registered residence with the authorities within 2 weeks of having moved in AND you must apply to the local immigration office (“Ausländerbehörde”) for your residence permit within the first 90 days of your stay in Germany.
You may only take up employment once you have been issued a residence permit explicitly authorising such employment.
Based on the information from the German Embassy in London, processing times for national visas can vary from about 2 weeks for some categories of employment to up to six months for self-employed professionals, which means that an application must be submitted well in advance.
My suggestion is therefore that you contact the German Embassy to get as much information as possible based on your circumstances.
I hope this information is helpful and I wish you the best.
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