I actually have two ideas for this type of crossover, and instead of separating them into two parts, I decided to put them both in this request. They won't be combined together unless you decide to go that route with the ideas. I'll try to add a paragraph or two detailing what they would look like combined, since the ideas share similar ideas and would, in theory, work well together or apart.
My first idea is for Archer to be mysteriously transported to the world of "Magical Index/Scientific Railgun/Accelerator". I want his contract with Alaya also mysteriously severed, but the power he's acquired from Alaya to stay. I then want him to think at first that the various characters are using this world's version of magecraft, which he finds to not be entirely untrue, but also not completely correct.
He finds out, after digging a little deeper into how things work in the city, that they all have lower levels of reality marbles, but without chants accompanying them. He discovers that the origin of reality marbles themselves appears to have come come from this world, rather than from the Nasuverse. Say that his level is immeasurable, and have everyone just think he's a level 1. Have Archer go along with this idea, before revealing himself during one of the first major fights of the series. Have him interact with the other main characters prior to this reveal, just don't have him reveal how powerful he is until the major fight.
I chose Archer to be the protagonist rather than Shirou, because I don't really want his heroic tendencies interfering with the story too much. He'd have all of the spotlight, and that's not the point of this idea. Instead, I want this story to go similarly to how the canon events play out at first, before eventually going off the rails in completely different directions that the story doesn't usually go in.
My second idea is for Archer to have been born in academy city in the first place, and not have him have been transported to there. Under these circumstances, you'd have to make him an already established level 6, which I feel might make the plot difficult or broken, or you'd have to remove his Unlimited Blade Works reality marble, and only allow him to use the swords themselves, not the world they come from.
In this latter case, you could perhaps introduce Unlimited Blade Works later on, assuming you make Archer reach level 6 by that point. For this idea to work, you'd also need to remove the whole thing about him seeing weapons and being able to copy them. Instead, you'd probably need to have it so that he doesn't know where the swords come from, and also have it so that he has to train with each individual sword that randomly appears.
In other words, he doesn't know where the sword come from, he just knows that he has a huge arsenal of weapons, not all of which he can access, particularly the higher level ones. New weapons can appear in the situations that demand them. Archer won't be able to structurally analyze, reinforce things, or create broken noble phantasms, unless you just say that those are a part of his powers somehow. You're basically going to need to majorly nerf Archer if you go with this idea.
Since this Archer hasn't experienced lifetimes of personal mental anguish and suffering, make him have slightly heroic tendencies like Shirou, but only slightly. Make him more agreeable and less ruthless, too.
An alternative idea is to make him be an immeasurable level of immeasurable potential. This would allow for all sorts of things to fly by without disrupting things too badly.
If you combine the ideas, I'd rather you try to not nerf Archer as much as possible, so maybe go with the whole immeasurable level idea, and allow him to use Unlimited Blade Works, but only in situations that require it. Alternatively, you can go with a situation in which Archer from the "Magical/Scientific" world gets the memories and experiences of Archer from the Nasuverse. If you go with this idea, I'm not really sure where you'd go from there, but I'm sure it would be very interesting to look at.