Towards the end of The Magician’s Nephew, the first king and queen turn out to be:
Frank, a London hansom cab driver, and his wife Helen.
They were transported from Edwardian London, so they would both have been familiar with 1900s technology, including mechanical sewing machines. (Especially the queen, who had been brought up in an era when women of her social class made clothes for their family members, or at least repaired or adjusted bought or hand-me-down clothes.) At some point the king and queen are going to need new clothes (especially since they didn’t get a chance to pack), and it makes sense that they would mention the convenient labour-saving device to help.
So who makes the sewing machines? The dwarfs are the most likely candidates (bold added):
“Carry him aside and lay him down,” said Aslan. “Now, Dwarfs! Show
your smith-craft. Let me see you make two crowns for your King and
More Dwarfs than you could dream of rushed forward to the Golden Tree.
They had all its leaves stripped off, and some of its branches torn
off too, before you could say Jack Robinson. And now the children
could see that it did not merely look golden but was of real, soft
gold. It had of course sprung from the place where the sovereigns had
fallen out of Uncle Andrew’s pocket when he was turned upside down;
just as the silver tree had grown up from the half-crowns. From
nowhere, as it seemed, piles of dry brushwood for fuel, a little
anvil, hammers, tongs, and bellows were produced. Next moment (how
those Dwarfs loved their work) the fire was blazing, the bellows
were roaring, the gold was melting, the hammers were clinking. Two
Moles, whom Aslan had set to dig (which was what they liked best)
earlier in the day, poured out a pile of precious stones at the
Dwarfs’ feet. Under the clever fingers of the little smiths two crowns
took shape—not ugly, heavy things like modern European crowns, but
light, delicate, beautifully shaped circlets that you could really
wear and look nicer by wearing. The King’s was set with rubies and the
Queen’s with emeralds.
It’s reasonable to assume that the talented dwarfs just needed an overview from the king or queen to be able to work out a design. They may even have developed a small industry with the moles.