Firstly it's Braunschweig, not Brunswick!
So on topic, I actually live close to Braunschweig, so I know the campsites I'm about to mention, but never actually stayed at them.
Firstly about an hour West of Braunschweig, is
Steinhuder Meer, a really pleasant spot with some campsites dotted around. Closer to Hannover you have
Birkensee campsite, from what I've heard it's pleasant enough with a small lake if it's hot, which it has been lately. I'm not sure Braunschweig itself has a campsite,
but Wolfsburg does. Wolfsburg is just 20 minutes East of Braunschweig and the home of Volkswagen. The site seems a little quirky, but again it's next to a Lake, the Allersee. The Allersee is perfect for a dip if it's hot, and a really pleasant place to walk around if it's not so warm. I was swimming in there just last week. It has sandy beaches and good amenities.
Many of the sites around here are a little tired and the Germans running them can be a little odd, but that's Northern Germany for you. There are all in the main, clean and functional places. I've heard rumour of a good campsite here in Celle as well, but I don't know where. I'll have a look later. I really
don't recommend the Silbersee in Celle though, a lovely place to take a dip, which I do quite often, but the facilities are terrible. I really wouldn't want to stay there.
I'll update later if any more springs to mind.