Could the greetings have been published in Melody Maker?
Since the subject of these recordings were greetings from Whiteman band members to the Orchestra's fans in England I would think it possible that that when Edgar Jackson went back to London he would have listened to the discs and had a secretary transcribe what was on the discs and had that published in Melody Maker.
I would think that Edgar Jackson, because he was not just any fan but the editor of the Melody Maker, would want these messages by Whiteman band members to British fans published in the magazine.
To me that would be a possible reason to have the recordings made in the first place. So the records could be played back in England and the messages published in Melody Maker.
Also admittedly a long shot but since the discs lasted until 1941 there were home recording machines in common use in the late 1930's and early 1940's couldn't it have been possible that one person Edgar Jackson played the recordings for have made a dub of them for himself?
I just think someone would have made a copy of the original discs because of their historical importance. As was mentioned the discs must have been played occasionally and someone might have made a home recorded copy of them.