Hello, I'm TV Tech1 and I'll be working with you to identify and help you resolve your problem.
I'm fairly confident the HDMI cable you own is at least an HDMI 1.3a cable, but if its a cable from the mid 2000's it can be the reason you get no sound at all from the Skybox.
The issue might be the Audio output setting on the Sky HD box, but if that were set to Stereo instead of Surround, I would expect you to have at least stereo sound.
The HDMI cable has to be the correct version to deliver Surround to the TV set.
Then....the TV should automatically select stereo or surround output...even though the speakers are stereo, not surround sound based.
So...I think the first thing I need to ask is this....do you at least get plain old boring 2 channel stereo on the TV speakers, or are they dead as a doornail?
Second, I should warn you that TV stations and TV shows are still pretty much stereo audio. And I don't think that surround audio is going to catch on with TV shows as much. The surround audio setup is used more in Movies and especially on DVD movies. Even more, you'll find Surround used quite well with Animated movies because they can create a 3 dimensional sound stage with animation, and the real world with real action doesn't support a 3 dimensional sound reference as well.
We hear in 3 dimensions, but recording in 3 dimensions....it's more subtle and harder to control during production.
Anyways, I mention that just to let you know that TV shows may never have surround sound due to that complexity.
Since you stated --
When my Sky HD box is set to send Dolby Digital over the HDMI cable (which is absolutely what I want), the TVs play no sound at all through their speakers
it does make me wonder if the TV at least receives stereo sound when the Skybox is set to stereo instead of surround.