Working for a UK company while full time resident of Germany

Please read the various other discussions of the same topic that took place on this forum in the last few months. There seem to be many Brits now who try to stay inside the EU at the last moment.
Assuming after Brexit you will be treated like any other non-EU-citizen, you will need a visa that allows long-term residence and work in Germany. A work visa (for being employed by the German entity of your employer) is one possibility, family reunion to your spouse (married or in a registered partnership) the other.
If you are not working for a German entity of your UK employer, you will be counted as self-employed, which will complicate matters.

I would point out that according to German law, an employer pays half of the health insurance cost but this is NOT a group insurance over the employer like in the USA. The employer usually has nothing to do with other Social security payments. And an employer certainly doesn't pay one's income taxes.

But like Beppi mentioned. If you are being paid from a company abroad without it being registered in Germany then you will likely be seen as self-employed. In this case the employer has no legal (but possibly contractual) obligation towards your health care costs. But since you will be residing in Germany then you almost certainly have to pay German taxes. It is strongly advised that one go to a tax advisor known as a Steuerberater.

If your permission to work in Germany comes into question because of Brexit then it is advisable to go ahead and get married. The permission one would have through a family reunion visa includes being able to work but also has some requirements. Usually you will have to pass an A1 German exam for example.