'Seven swans a-swimming

Six geese a-laying

Five gold rings!'

Nigel Channing sighed. 'Figment, must you sing that dreadful song over and over again?'

'I thought you liked my singing, Doc,' said Figment, the little purple dragon.

'It's bad enough I have to hear One Little Spark over and over -'

'I thought you liked the Sherman Brothers.'

'I have nothing against them.'

'But you don't like the Twelve Days of Christmas.'

'It's the 4th of December, so not actually the time for the song,' said Nigel.

'But it is, Doc!' said Figment, happily. 'It's always time for Christmas carols at Christmas!'

'Figment, do you know when the first day of Christmas is?' asked Nigel.

Figment beamed. 'Eleven days before Christmas Day,' he said confidently.

'No, it's actually Christmas Day,' said Nigel.

'So all you get for Christmas is a partridge in a pear tree, but on the…' Figment muttered to himself as he counted on his fingers. 'On the 6th of January, you get twelve drummers drumming, eleven pipers piping, ten lords a-leaping -'

'Yes, Figment, I know the words,' said Nigel.

Out of nowhere, Figment pulled out a calculator and began to type furiously on it.

'Where did you get that?' asked Nigel.

'From my imagination,' said Figment. 'Doc, you only get one present on Christmas Day.'

'That's fine, I only want one present anyway.'

'But you get 364 presents on the last day of Christmas.'

'You're talking about the song.'

'Forget all the birds and jewellery you wouldn't have enough fingers for, thirty lords a-leaping would cause so much trouble on their own.'

'They're lords. I don't think they'd cause that much trouble.'

'But they're a-leaping, Doc!' said Figment. 'And besides, it's way too many gifts for January 6th!' Figment wailed, and the calculator poofed out of existence. 'Why January 6th anyway?' he asked, much more calmly.

'Because December 25th is when Christmas actually begins,' said Nigel. 'And it ends on the 6th of January.'

'Why?' asked Figment.

'Well, according to the Bible, the Three Wise Men gave their gifts of gold, frankincense, and myrrh to the baby Jesus, twelve days after Jesus was born,' Nigel explained. 'It's known as Epiphany, or Three Kings Day. Sometimes celebrated as Twelfth Night.'

'Like the Shakespeare play,' said Figment. 'But didn't they celebrate the new year from BCE one to CE one?'

'I don't think so, Figment, celebrating the New Year in January was a Roman thing and the Jews didn't very much like the Romans at the time. Besides, we as a society only started celebrating the New Year on January 1st when the Gregorian calendar was adopted in the sixteenth century.'

'Wow. That… wasn't that long ago,' said Figment.

'No, it wasn't,' Nigel agreed.

'So, because it's 2022, the modern day, if I was to get all these presents from my true love, then on New Year's Day 2023 I'd get seven swans a-swimming

Six geese a-laying

Five gold rings!'

Figment resumed singing and Nigel put his head in his hands.

I picked Seven Swans a-Swimming for Figment to be singing randomly when the story picks up, but I did the calculations and Figment is right - the gift the narrator would receive for New Year's Day would indeed be Seven Swans a-Swimming. What a spooky co-incidence.
Everything said is true - the first a day of Christmas is Christmas Day. The period runs until January 6th, the day the
wise men (or kings? I'm not sure), give their their gifts to Jesus. It's never stated how many wise men there were but common assumption is three.
In the song, for Christmas day itself, you're only getting one partridge and one pear tree. But by the end, the 6th January, you're getting 78 gifts. Plus all the rest you accumulated, so eventually, you're left with 364 gifts. Most of them flocks of birds. Hope you like birds! But you're also getting fifty gold rings and twelve pear trees too, so it's not a total loss.
Celebrating New Year in January was a thing the Romans did. Jewish people had (and have) their own calendar, different to the Romans so they would have different celebrations. Also Jesus was likely born in summer so there would be no New Year's celebrations during the time of his birth in the Bible.
And the Gregorian calendar was standardised the world over in 1582. Meaning New Year's Day as a celebration is younger than the painting on the ceiling in the Sistine Chapel.
Though other cultures and countries around the world celebrate on different dates and months worldwide. For example, Halloween started off as the Celtic New Year.