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Not just document, but also start raising it up the command chain.

So as the days go by without a response - you pass a message to your management that you've had no response. And you escalate the tone of those messages - so a few days beforehand, you are telling your management that you will not be able to go live "because this other department are not co-operating".

That way, you have evidence that you did all you could to make things happen, AND you can document that your management knew there was a problem, AND you can document that you did everything within your power/authority to make things happen. If the brown stuff does hit the air movement device, then you should be out of the firing line - or at least you've made yourself a good umbrella. But hopefully, if your management is any good, they'll have made things happen and the problem will have been resolved before that happens.

It's not mentioned in the article, but it's also good to make sure that in your request it is clear what the timescale is - you don't want the other side to be able to use the excuse that they weren't aware of when it was needed/it was urgent/whatever.