This isn't British, and it isn't slang. In general, most English-speakers don't worry about blasphemy these days.
'God damn' would be an odd thing to say in this context though. On its own, it isn't a common expletive.
You could use 'god damn(ed) ' as an adjective, as in 'It's raining again and the goddamned tent pole has broken'
There is 'God damn it' as an expletive, as in 'God damn it! It's raining again', or you could use either 'God!' or 'Damn!' on its own to show annoyance, disbelief or frustration. Nobody would be offended.