This is outrageous, and the more I think about it, the more wondrous it gets
For all intent and purpose, the electromechanical relay was invented by Joseph Henry in the late 1820s. If that name sounds familiar, it should. The SI unit for magnetic inductance was named the Henry in his honor. But what's amazing is that electromagnetic relays were not used for computing for a century! Heck, you didn't see them in telegraph machines until the 1860s.
But! What if enamel-coated wire could be manufactured to a smaller gauge and more economically!
It was the height of the Industrial Revolution! Factories were popping up everywhere! If Hollywood can be believed, Dr. James Moriarty is poised to take over the world! And all someone had to do was draw that magic line between the analytical functionality of Babbage's mechanical wonder — and a relay that would take up a fraction of the volume.
Boom! Electromechanical computing in the mid 1800s.
But, why is that important?
Because the most common reason any technological advance took place when it did is that there was a reason to look for it. This is really important! In many cases, it's not the technology part of the tree that's causing an advancement to take place when it did, it was the fact that a need for it finally arose. Necessity is quite often the mother of invention.
And if we have serious computing going on in the 1800s, we have the need for memory in the 1800s (not just registers, but serious computational memory on the order of whole Kilobytes!) Electomagnetic relay computing would do it, but it's also important to understand why.
Babbage's mechanical wonders had memory. Persistent memory. If you stopped turning the proverbial crank, all the gears stopped where they were and whatever had been stored would be remembered — forever. Why would anyone want memory that would, eventually, degrade?
Because you can only get so much out of spinning gears. Relays are fast! Lightening fast! And the best way to take advantage of that speed is to have memory that's just as fast as the relays are!
The necessary technological advances that would bring about magnetic core memory already existed in the mid 1800s. Electricity and magnetism. What lacked was a reason to even think about the need for this marvelous thing called "memory." And that's what we've provided by bringing electromagnetic computing to the fore a century before it did.
But even this requires a reason. Maybe. That's the wonderful thing about stories. You don't really need a reason for anything. But you might want to consider why you need faster computation in the mid 1800s. It's not like they were putting people into orbit (seriously, go watch the movie Hidden Figures or read the history of those amazing ladies. NASA didn't need computers as we know them today for quite some time, because human computers were fast enough. It wasn't until they needed the data faster that they started replacing the people...). What in your story needs fast computing? If you answer that, you've justified electromagnetic relay computers, which justifies researching memory, which brings MCM into the fore a century early.
In my comments I mention that MCM likely couldn't have been brought to light more than about 15 years before it did. This is because of this dependency on the need for speed. MCM was dependent on fast computation. Bring about an early reason for fast computation and you speed up a lot of things.