I remember listening to a "Just William" audiobook when I was younger about a Christmas Day when he received a mouth organ, and started playing it really early in the morning.
‘The Christmas Truce’ by Richmal Crompton, first published in William’s Happy Days (1930). It takes place at Christmas:
‘Him?’ he exploded fiercely when at last the power of speech returned to him. ‘Ask him to my Christmas party? I’d sooner not have a Christmas party at all than ask him to it. Him! Why I wun’t go to the King’s Christmas party, if he was going to be there. Not if I had to be beheaded for it.’
William receives a mouth organ:
Bates took the parcels out one by one, announcing the name on each label. The first was William. The Hubert Laneites watched him go up to receive it in paroxysms of silent mirth. William took it and opened it, wearing a sphinx-like expression. It was the most magnificent mouth organ that he had ever seen. The mouths of the Hubert Laneites dropped open in horror and amazement. It was evidently the present that Hubert had destined for himself.
William plays the mouth organ, though the story does not say at what time of day this takes place:
The Outlaws marched triumphantly down the road, brandishing their spoils. William was playing on his mouth organ, Ginger was flashing his electric light, Henry waving his fountain pen, and Douglas slashing at the hedge with his penknife.
(This was easily found using a Google Books search for “just william mouth organ christmas”.)
Update: The incident whereby William is “the first person up on Christmas Day … and wakes everybody up” comes from the story ‘A Busy Day’, first published in More William (1922):
William awoke and rubbed his eyes. It was Christmas Day — the day to which he had looked forward with mingled feelings for twelve months. […]
Singing ‘Christians Awake!’ at the top of his lusty young voice, he went along the landing, putting his gifts outside the doors of his family, and pausing to yell ‘Happy Christmas’ as he did so. From within he was greeted in each case by muffled groans.
He went downstairs into the hall, still singing. It was earlier than he thought — just five o’clock.
I submit that you have mingled these two stories in your memory! This would be easy to do as both stories appear on the Just William at Christmas audiobook, read by Martin Jarvis.