It's been a while since I read it but Lynn Olson and Stanley Cloud's For Your Freedom and Ours is a good account (it has a different title in the US, I think).

Their book focuses on the pilots of what would become 303 (Polish) Squadron, but also talks about some of the other stories. It's a cracking read, and I'd heartily recommend it. Unfortunately I lent my copy to someone and never got it back so this is from memory. Hopefully someone can expand on the detail.

There were a number of different sources of Polish personnel in the West. The first wave, as you might expect, was of those escaping from the German/Soviet attack in 1939. One of the Polish government's strategies was the so-called Romanian Bridgehead, in which troops would fall back to the Polish-Romanian border (the two shared a short border in the far south-east of Poland around what is now Zalishchyky, Ternopil Oblast, Ukraine) and hold the passes while supplies were brought in through Romania, which was allied to Poland at the time.

Unfortunately, the Romanians, under pressure from Germany and with their own problems, elected to declare neutrality and intern upwards of 100,000 Polish troops crossing the border. However, between a certain amount of official laxity (letting the Poles out on day release to help with the harvest, for instance), much bribery at the local level and plausible deniability provided by the Polish embassy in Bucharest running a huge fake-papers operation, thousands of Polish servicemen made their way in ones and twos, with papers proclaiming them to be everything from students to priests, to Black Sea ports like Constanta, where they were taken by neutral ships (either bribed in advance by the Polish mission or by the men on the spot), usually to the French Levant. (Although the men were told to carry nothing that could officially link them to the Polish military, most kept their wings, or ceremonial daggers. One schlepped the two-bladed wooden propeller of his crashed biplane all the way through Romania, via Syria and France, to the UK, where it still stands in the lobby of the Sikorski Institute in London)

That was the route taken by most of the Kosciuszko Squadron guys, so is what the book focusses on but other groups managed to make their way to the Baltic coast and out to Sweden, or from Romania into Yugoslavia and then Italy. These men formed the bulk of the Polish forces in France and the UK in 1939-40. The exploits of the pilots are well known but there were also Polish infantry divisions engaged in the Battle of France.

The second major group were those taken prisoner by the Soviet Union in 1939. While many - officers, intellectuals and so on - were murdered at Katyn soon after capture, most were sent to labour or concentration camps in Siberia. This became something of an embarrassment when the USSR found itself allied to Britain in 1942 and after some nagging, they finally agreed to release them.

As in literally, they opened the doors and said. "War's that way. Good luck", leaving the Poles to trudge all the way south through the Caucasus and Iran (where some of the men adopted a bear cub they found being fed by a little boy at the roadside) to the Middle East, where they formed the core of II Polish Corps, which fought in North Africa and Italy, most famously at the Battle of Monte Cassino, where the bear, now fully grown, carried ammunition for the artillery.

Finally, as you say, many Poles were pressed into service in the Wehrmacht's Ostbatallions and deserted or were captured in the fighting in Normandy and France. These men were by and large happy enough to join up with the Allied forces and fight Germany instead.