had to explain the meaning to my *young driving companion.

Then you had to explain to the board who your *young driving companion was

Not my grandchild, smarty pants! And never you mind!

"Mum" being for employers; "ma'am" being reserved for The Queen.

OK. This means the Queen Mum employs the Queen?

I was taught you say "Your Majesty" first then "ma'am" in rest of conversation.
Ah HA!...Debrett's "Etiquette and Modern Manners" says :
The Queen : Correspondence should be addressed to "The Private Secretary to Her Majesty The Queen.
Introduced as : Her Majesty The Queen.
Addressed as Your Majesty subsequently "Ma'am" (pronounced mam)
The correspondence for Queen Mother same form as for the Queen.
Introduced : Her Majesty, Queen Elizabeth, The Queen Mother
Address : Her Majesty subsequently Ma'am.

I was under impression the British people started calling Her Majesty, Queen Elizabeth, The Queen Mother by the cozier and most affectionate soubriquet "Queen Mum." When in Britain I noticed Mothers were most often called Mum by children. Can/will any Brits help me out here?

Now, if you are talking Mothers and Daughters ......