Every now and then I hear about famous Polish, Czech, French, etc pilots and troops that escaped to allied countries and served later in the war. How did they get across a hostile Nazi occupied Europe? Were they stationed in the remote enough places to escape before the Nazis got there?
There were many governments that escaped the country that they were from simply be leaving right before it fell, thus creating multiple governments in exile. Sometimes, troops and civilians would escape with them. Most of the time, though, those left behind would simply leave and go to a friendly nation.
Poland is a good example. The Polish government retreated to France after Poland fell, and set up a headquarters, so to speak, in Paris. Civilians and soldiers escaped via boat, plane, or by going through other countries, as is the case with this soldier here, who escaped through Hungary, via Budapest, Italy, and finally, France. Many refugees east of Germany, like Czechs and Poles, also escaped to the east to Russia, or north, reaching Sweden, and then going down to France or England.
What would end up happening (at least in the case on the western front), is that there would be a large number of foreign governments and "armies" (albeit not large enough to constitute a superior force) around France when Germany finally invaded. At this point, the Germans would beat back the Allies to the coast, causing "Operation Dynamo", the evacuation of Dunkirk. This would be the sight where nearly 340,000 troops would be evacuated, among them a large number of French, with a number of Dutch, Belgians, and other foreign nationals, such as Poles and Czechs, as seen here. Other instances, like the French Submarine Surcouf, simply steamed to British ports and either surrendered, or decided to join the fight against Germany.
Once in the U.K., the remaining troops and governments in exile would form new armies (along with British help), or volunteer to fight in British units. Pilots would either volunteer to serve, such is the case with the Czechs in joining the RAF, or formed their own squadron, like the Polish 303 Squadron, while former soldiers and civilians volunteered to serve in the new armies, like the Free French Forces, who would also draw from volunteers in French colonies, or the "Polish Armed Forces in the West". Ironically, in many of these cases, some of these troops would be better trained and have more experience than the countries they would reside in.
I hope this helps answer your questions! If you have any more, please feel free to ask!