Pondering this question and I can't seem to find any real information on it.
Once war was officially declared, what happened to diplomatic staff in affected countries? For instance, were diplomats stationed in the British embassy in Berlin allowed leave and return to the UK? (And vice versa). Further, what happened to American diplomats stationed in Japan upon the attacks on pearl harbour?
Moreover, during the war period, could diplomatic staff travel freely between hostile nations? For instance, was there terms set so a British diplomat could hypothetically travel to Berlin on official business to discuss the war between the 2 countries?
I can't answer for every nation but since you mention the British Embassy in Berlin I will provide an answer for that. According to the book Outbreak 1939 they were repatriated in a somewhat strange calm. The book points out that, half an hour after Britain's ultimatum to Germany expired (placing the two at war), the British ambassador was requested for an interview with Ribbentrop. By the end of the day they were in the Adlon Hotel waiting to be sent home, drinking with American Journalists. The Ambassador, Sir Nevile Henderson, returns from the interview (at which he asked for Germany to abide by the Geneva convention) and returns to the Embassy where 'he is surprised to see that the telephone lines are still working'. His last mention states he returned to the UK via Holland. There is less information on German diplomats but what there is shows a similar type of very calm evacuation(the book was published by the IWM so go figure)