The German occupation of Prague did not stop all travel in and out of the city or travel to other countries. For example, the head of the American embassy in Prague at the time of the invasion, Envoy Wilbur Carr (he did not have the title "ambassador"), remained in Prague for another month before returning to Washington after the State Department determined that there was no point for him to be there anymore. The now more famous George Keenan, a junior member of that legation at the time, actually stayed behind to look after the American property, and he only left after the start of the Second World War. Flights remained open between Prague and Paris, which was the preferred destination for political exiles leaving Prague, as France acted as a hub from which a person could go anywhere.

Nevertheless, there were those who needed to flee Prague during the Nazi occupation because of their political ties to the pre-Munich, anti-German governments of Thomas Masaryk and Edward Benes. Emanuel Voska, a wealthy Czech-American who had made his fortunes as an immigrant in Chicago before becoming intimately involved with Czechoslovak founding father Thomas Masaryk during the First World War, described one such escape attempt in his memoir Spy and Counterspy, which he wrote immediately upon escaping Prague and arriving in New York. According to Voska, as soon as he heard that an invasion was underway, he drove to the Prague airport in order to fly out of the country. Long before troops arrived in Prague, however, German planes had already occupied a position over Prague, preventing Voska's escape (and anyone else's). Voska returned home to make different plans to leave the country, but Nazi troops had already begun occupying the city by the time he got there. The Germans quickly arrested the high-profile Voska, though after some amount of time (it's not clear in my notes, and unfortunately I don't have the book in front of me) they released Voska on medical leave (Voska claimed in his memoirs that he tricked his captors by eating moldy bread, which led doctors to believe that he had a severe gall-bladder attack). Voska then flew, legally, to Paris, and then he traveled to the United States (note that Edward Benes had already fled to the United States by this point in time and had taken on a prestigious position at the University of Chicago, making the United States the preferred destination for Masaryk's old inner circle).

I am afraid I cannot help you with a precise timetable of events on that particular date, nor a comprehensive overview of the possibility of fleeing the country. However, I can attempt to answer a very small part of your question with a particular focus on the Jewish inhabitants of Czechoslovakia. Hopefully, despite these limitations it will be of some interest to you.

The fact is that fleeing Czechoslovakia, or the soon to be Protectorate of Bohemia and Moravia, would have been possible through both legal and illegal means, although not easy, for some time. So one would not necessarily have to outrun the Wehrmacht on March 15, so to speak.

After the occupation in March, 19 016 Jews legally left the Protectorate in 1939, followed by 6176 in 1940. In the next three years, the numbers dropped significantly to a total of less than a thousand. Poland and Slovakia were their most likely initial destinations, but many would move on as the Germans expanded into Europe. Some five thousand would also leave ilegally. Roughly a fourth of its 118 thousand Jews are estimated to have left the rump state.

In addition to local Jews, several thousands of Jews who had fled Germany, Czechs and Germans from the Sudetenland, and Germans not sympathizing with Hitler's regime left in fear of persecution would also be looking to leave the country and share a similar fate as their Czechoslovak counterparts. The Czech Refugee Trust Fund founded by Britain in 1939 was specifically set up on helping these Germans and Jews from the Sudetenland, and only later would be used to help Czechoslovak refugees in general.

Similar to the 1938 establishment of the Central Agency for Jewish Emigration in Vienna, as well as in other European cities later on, German authorities took over passport, emigration and deportation duties in the Protectorate in July 1939. Jews could leave the country, indeed the Germans were actively attempting to drive them out at first, but they had to be processed by this agency, and were stripped of most assets. In October 1941, after Heinrich Himmler ordered Jewish emigration to be suspended – hence the aforementioned drop in numbers, as well as the thousands emigrating illegally – deportation and removal of Jews became the purview of this office.

Furthermore, fewer and fewer countries would accept the refugees, and not only because of the war. Already after the events of the Kristallnacht, the Anschluss or the Munich crisis many countries would enact stricter visa requirements, limiting potential destinations not only for Jews. Jewish immigration to Palestine was tightly controlled by the British, leaving to the United States would also soon pose a problem as no visa applications made after the establishment of the Protectorate were processed successfully. Those fleeing Czechoslovakia would then have to resort to illegal transports to Palestine or rather exotic destinations that would yet not turn them away such as far-off Shanghai or the Dominican Republic.