You can probably modify any cheese to melt similarly to velveeta.
Sodium citrate would be my go-to method. It's actually one of the main things that gives Velveeta its smooth, processed texture, but you can buy it yourself and add it to a cheese of your choice. (A good alternative would be gelatin plus (evaporated) milk; see rumtcho's answer for a description of that. It's not quite as close to processed cheese texture, but it should be good enough for your purposes.)
The basic process is to heat a bit of liquid (water, milk, broth, beer, whatever you like) with sodium citrate in a pan to dissolve it, then slowly blend in grated cheese, letting it melt, until it's all incorporated. Ideally, you use an immersion blender to get it really smooth, but I think determined whisking would suffice. For thinner sauces, you can use more liquid. I've also seen varying ratios of cheese to sodium citrate, anywhere from 25:1 to 90:1.
Here's an example generic recipe for just the cheese. I've seen this kind of thing used generally for sauces, for example this Serious Eats "Modern" Mac and Cheese recipe or this nacho cheese recipe, but it should work for you too as a melted processed cheese replacement.
That example recipe says to use immediately; it'll be like melted processed cheese at that point. If you let it cool, I'm pretty sure it'd solidify into something like Velveeta.
Note that if you want it to be really thick, you could likely use slightly less liquid, as long as it's enough to cover the bottom of your pot and dissolve the sodium citrate. You could probably use a little less sodium citrate too and still have it be plenty smooth while seeming a bit less processed, but since you're actually trying to replicate processed cheese, I guess you might as well go all in!