...how long would it really take for her voice to get to Betty's father's radio?
Radio waves travel at 300,000,000 meters per second (the speed of light). What Betty was hearing was happening in real time.
Why don't we put it to the test...by using the same type of radio that Amelia had, and going to where Betty lives or used to live, and using somewhat of the same setup and see if anyone could be heard from Niku!
As a Certified Broadcast Radio Engineer (or maybe that should read Certifiable!

) it would be near impossible to recreate this today. Just as there were many AM signals that had to be filtered through to hear the Electra in 1937 (not AM broadcast stations but amplitude modulated transmissions) not to mention the challenges of picking up a signal on a harmonic of the original carrier, we also have to deal with the propagation characteristics on those frequencies as well as the properties of the antennas, both on the plane and in Florida (or Rock Springs or elsewhere). Now fast forward to 2014. The number of RF signal generating devices from regular radio to appliances to computers to industrial applications to retail to cell phones.....it's grown a hundred-fold exponentially since I first got into this business over 30 years ago. What would have been considered a strong signal in 1938 from a radio station I used to work at (100 watts!) would be lost in the clutter and interference today. That station recently upgraded from 5000 to 10,000 watts just to stay in check with the electronic noise that has been creeping up over time.
Here's an easy experiment: Hop in your car, tune the radio to a decently strong AM signal in your driveway. Now drive to any nearby gas station and pull up next to the pump. What do you hear? The microprocessors in the gas pump radiating an RF signal in the form of noise. That's why it's referred to as the noise floor. In 1937 the noise floor worldwide was pretty much limited to fluorescent lights, automobile ignitions, power lines and thunderstorms. Add in today's electronic lifestyle and industry. I'm sure you can see why traveling to Niku and trying to experiment with a 50-watt AM transmitter on 3105 kHz to see what we would hear halfway around the world would prove unsuccessful. Not that that signal couldn't travel that far, but more to that we would not be able to pick it out of the noise floor.
Cool idea but unfortunately for our theory-proving purposes, the playing field has changed. Also, as I recall, there are no examples of the transmitter used in the Electra from which to try this experiment which is why Bob Brandenburg and others have so closely scrutinized the characteristics of that particular model of transmitter and the design of the antennas to be able to determine the validity of all the logged radio receptions reports involving the Electra.
Oh to have been a fly on the wall in Betty's (or Randolph's or Mabel's or the other few) homes when those messages came through from so far away! To observe Amelia's psychological perspective change as the entire chain of events played out would be a fascinating study all by itself.