I haven't seen the movie, and I don't know how Eric Lomax did it, but building a homemade radio is simple, and it used to be a very popular amateur electronics project. The key is that you need a vacuum tube.

Here is an example of such a project: http://www.instructables.com/id/Make-a-One-Tube-Radio/

And here is a page about all the radio projects that one man did over the years, including many that were based on designs from the 1940's: http://www.bignick.net/Morgan_Radio/Radio.htm You can see from the pictures that they were not very complicated, and those are commercial-grade radios.

A radio project would be even simpler if you didn't use amplification (which is not strictly necessary), although that would greatly limit the radio's range and would probably not have worked in this case.

The BBC was broadcasting in India at the time, so although it's still fairly impressive that the radio was able to get that kind of range (1000 miles or more from that part of Thailand to, say, Calcutta), it's not as bad as getting a broadcast all the way from England as you may have been thinking. Keep in mind that Lomax was an experienced telegrapher and radio operator, so he knew what he was doing; part of his military training would no doubt have included doing radio repairs in the field.

I don't know much about the history of radio, but in those days they may have been using different frequencies that traveled farther.