Solved it myself. I put the iD SIM in my iPhone and noticed that in Settings-Phone the number shown was still the previous SIM's number. I changed this to the iD number (which I eventually discovered in the email documentation sent when I purchased the SIM) and saved it. I then returned the SIM to my iPad - lo and behold, the Mobile Data Number now shows the correct phone number. I can only assume that the number in the SIM needed to be saved in this way to kick-start it, which option, of course, is not available in the iPad.
However, the Carrier and Data Plan are still showing Three. I know that iD is an MVNO using the Three Network - but that doesn't explain why I get different info (I think the correct info) for the Carrier and Data Plan on my iPhone. Anyway, I'm not too bothered about that; I was more concerned about accessing the SIM phone number easily.
My experience with this and a few different issues suggests that many carrier SIMs don't play too well with tablets. The industry seems oriented to phones and phone capabilities. For example when I registered my new SIM I was required to enter a security code which they sent - you've guessed it - to the SIM phone number which, of course, I could not receive in an iPad.
