“How can we ask our German colleagues to respond more quickly without annoying them?”
Explain to your German colleagues what time looks like in the specific situation. Lay out the cause-and-effect relationships, i.e. if late by this much time, then this happens.
Prepare, and inform them about, contingency measures you will need to implement in order to react to the negative consequence of lack of speed.
Ask your German colleagues if there is anything you can do to help them to move faster. Offer suggestions on how you might be helpful in speeding things up.
Finally, and very importantly, reflect on your need for a speedier response from your German colleagues. Is speediness truly important? Who and/or what is driving speed as a priority? Is it a real or a perceived need?
Is your customer – whether corporate-internal or -external – really demanding it? In fact, why not ask your customer? Do you have the courage to ask your customer what is truly important to them?
If you do not have the courage, why not? What kind of business relationship is it if you feel that you cannot ask such questions?
Perhaps there are good reasons for you to be patient and/or for you to ask your customer to be patient. Perhaps because you and they will receive a higher quality result from Germany. “Patience is golden” and “Haste makes waste.”
Finally, all too many Germans have experienced American colleagues who make all sorts of things urgent, only to find out later that they were not all that urgent. If your German colleagues experience false-urgency two or three times, they become very skeptical of any and all future urgent requests.