No earning income yet, but need healthcare soon (Frankfurt)

Dear all,

What do I need to do about healthcare, as I plan to register as living at my girlfriend’s address in Frankfurt next month?

MY SITUATION:

I moved to Frankfurt to be with girlfriend in July Protected content .
I'm a British citizen.
I’m still registered in the UK at my parents' home address and at my local GP/doctor's surgery.
I have a European Health Insurance Card.
I started a UK Limited company in May Protected content it is unlikely I will create any revenue until next year (so I am not earning any income at the moment and living off my savings).

THINGS I NEED ANSWERS FOR:

I want the same sort of cover as I get in the UK under the NHS but I have been told that when I register as living in Germany my European Health Insurance Card will not cover me.
I have been told when I register as living in Germany it is law to get health insurance, so I need to arrange this ASAP.

I hope you can help me clear this up, as my girlfriend/friends don’t know what I should do.

Many thanks!