The tuition fees at private universities in Germany range from a few hundred euros to several thousands of euros.
Yeah, there are a lot of public universities that teach engineering in English in Germany.
First and foremost, if you are going to study at a university in German, then you must prove that you have sufficient knowledge of German at the Embassy to be granted a student visa. No university will give you an admission letter to study in German if you don't understand German.
Also, you can't just apply to a language course for instance like A1 and expect to be granted a language visa considering that there is a Goethe Institute in Nigeria that also offers A1. It is advisable to study German in Nigeria to a respectable level like B1 before applying for a language course in Germany.
Normally, most language courses are part of the requirement to pursue a degree program in German. Hence, once you have successfully completed the German language course, you can go ahead and start with your degree program and apply for the switch from language visa to a student visa in Germany. You don't need to return to Nigeria to apply all over again.