Do not send a fax but call them.

My experience with the consular services of the German embassy in London is very good and they should be able to produce one of the green (temporary) passports that you mention while you're there. Yes, they're for urgent travel like the one you're about to do and, yes, they can be produced immediately.

It won't be cheap, though, and you will have to travel to London unless they've changed things since I moved back to Germany. Note that there are also honorary consuls throughout the UK where you can fill in passport applications. You will have to ask the consular services in London, but these honorary consuls might also be able to help you with the green passport.

Since you only noted now that your passport has expired, you probably do not travel too much. But in general for people finding this thread via search machines one should note that

a) you are allowed to own TWO German passports, provided your employer gives you a reason for it (e.g., frequent travel to places where a visa is required). Having two passports reduces the probability of having no ID for traveling, but note that the 2nd passport is only valid for five years and costs the same as a primary passport.

b) if you moved to the UK sometime in the past 10 years you might still have a valid Personalausweis, which is perfectly good for travel within Europe. Since you're a permanent resident in the UK you've probably no Personalausweis, though, since German citizens who are not residents of Germany will not get one.