I've been in Berlin for a long time and it's a great place to be if you don't speak much German.
Your first stop would be the Berlin board of toytown Germany
toytowngermany.com
Toytown has a section on family life in Berlin, with a toddler meet-up group, and you can ask advice on where to find English-speaking doctors etc.
There's the Berlin Women's International Club:
www.biwc.de/
More American than British is this one:
www.awcberlin.org/
I was sure there used to be a British Women's Club 10 years ago but I can't find anything about it now.
If you're into church stuff there's St George in Westend, which also has an English-language breastfeeding circle, for instance. There's an American Catholic church in Dahlem, can't remember name.
If you want to put the baby into a nursery when s/he gets older (3 is normal here), there are plenty of choices for bilingual Kitas. We chose the Tom Sawyer Kinderhaus in Dahlem, which is Montessori and very modern. The JFK nursery is also popular. You need to register them a long while in advance, so looking round for places will give you something to do.
Although if you're forces you might not be in Berlin for long anyway. I haven't seen army people here for a long time - will you be up in Spandau? Spandau is a long way from anywhere, unfortunately, but there's a good underground into town.