Voldemort refers to himself in the third person (never a good sign, avoid people who do so) as 'Lord Voldemort':
Harry’s voice was high, clear, and cold, his wand held in front of him
by a long-fingered white hand. The man at whom he was pointing was
suspended upside down in midair... “I have it not, I have it no more!
It was, many years ago, stolen from me!” “Do not lie to Lord
Voldemort, Gregorovitch. He knows. . . . He always knows.” ~Harry Potter and the Deathly Hallows - The Thief
His followers, sympathizers (and Snape) refer to him as 'the Dark Lord' when speaking about him and in his presence they address him as 'My Lord'
“My Lord,” Yaxley went on, “Dawlish believes an entire party of Aurors
will be used to transfer the boy —”~Harry Potter and the Deathly Hallows - The Dark Lord Ascending
Bellatrix is the one who objects when Harry calls him simply 'Voldemort':
“Shut your mouth!” Bellatrix shrieked. “You dare speak his name with
your unworthy lips, you dare besmirch it with your half-blood’s
tongue, you dare—” ~Order of the Phoenix, chapter 35 -Beyond the
Veil
The people who fought him, like Dumbledore and Harry call him 'Tom' or ' Riddle' to his face, (which he hates as this name represents his muggle father) and 'Voldemort' when he is not present:
“So, Tom . . . to what do I owe the pleasure?”
Voldemort did not answer at once, but merely sipped his wine. “They
do not call me ‘Tom’ anymore,” he said. “These days, I am known as
—” “I know what you are known as,” said Dumbledore, smiling
pleasantly. “But to me, I’m afraid, you will always be Tom Riddle.
~Harry Potter and the Half-Blood Prince- Lord Voldemort's Request
and Harry in the final battle:
I’ve done what my mother did. They’re protected from you. Haven’t you
noticed how none of the spells you put on them are binding? You can’t
torture them. You can’t touch them. You don’t learn from your
mistakes, Riddle, do you?” “You dare —” “Yes, I dare,” said Harry.
“I know things you don’t know,
Tom Riddle. I know lots of important things that you don’t. Want to hear some, before you make another big mistake? ~Harry Potter and
the Deathly Hallows -The Flaw in the Plan
Other Order members also call him Voldemort. All the rest call him 'You-Know-Who' or 'He-Who-Must-Not-Be-Named'. As the only people dare calling him 'Voldemort' were the ones who fought him, he used a spell to track people with it:
“but the name’s been jinxed, Harry, that’s how they track people!
Using his name breaks protective enchantments, it causes some kind of
magical disturbance — it’s how they found us in Tottenham Court Road!”
“Because we used his name?” “Exactly! You’ve got to give them credit,
it makes sense. It was only people who were serious about standing up
to him, like Dumbledore, who ever dared use it. Now they’ve put a
Taboo on it, anyone who says it is trackable — quick-and-easy way to
find Order members! They nearly got Kingsley — ~Harry Potter and the
Deathly Hallows - Xenophilius Lovegood
But that doesn't mean he objects people uttering to the name "Voldemort". He doesn't seem to have a problem with it:
How did you escape with nothing but a scar, while Lord Voldemort’s
powers were destroyed?” There was an odd red gleam in his hungry eyes
now. “Why do you care how I escaped?” said Harry slowly. “Voldemort
was after your time. . . .” “Voldemort,” said Riddle softly, “
is my past, present, and future, Harry Potter. . . .” ~Harry Potter and the Chamber of Secrets - The Heir of Slytherin
I am not a big fan of the Cursed Child, but the scene you're referring to happens in a dystopian society, where Voldemort had won. People supposed to idolize him so 'you-know-who' is out of the question. There is no conflict of dark vs. light any more, so no reason calling him the 'Dark Lord'.
'For Voldemort and Valour' is an catchy alliteration so probably the guy Voldemort put in charge of the PR came up with it and it reminds both of Mao, Stalin and Kim Il sung's cult of personality and of the greeting in the Third Reich, so it seems convincing and fitting considering the situation.
