This is my understanding, but obviously get proper advice before taking a big decision.
As far as I know, Germany allows dual nationality by birth, and pretty much always has. (Historically, Germany has been very strict about taking on German nationality for people that do not wish to renounce their original nationality.) Currently, Germany also allows its nationals to take on the nationality of other EU countries without renouncing German citizenship.
I would assume that your child, born in Germany to a German father, would be a German citizen. That in itself would not cost anything, but it always costs money to get the documents (like a passport, national ID) to prove it. Ask around in Germany what documents it's usual to get for a newborn -- I have no idea. You might not need anything if you aren't traveling.
I would also assume that your child would be a British national through you. You need to check the consulate's website for your area in Germany and see what documents you need to provide/obtain. The countries I have experience with (US and Germany) have something like a birth certificate that can be issued by the consulate abroad that serves as proof that the child is a citizen by birth.
Why would you not get a British passport? Obviously check, but most countries don't like it when a national of the country tries to enter on a passport from another country. I don't know how strict the UK is but I know it is frowned on.
DD has 2 passports and we always have to bring both -- one to enter the US and one to reenter Europe. Now we have all got British nationality so DH and I will need to carry 2, and DD (and DC2, when born) will need 3 apiece!
Good luck sorting it all out!