Here's good pictures of a nicer than usual Automag IV in .45 Winchester Magnum, with a gorgeous polished finish and custom wood grips. This one belongs to a man in France, and I know of at least one Automag III chambered for .30 Carbine in Norway, so these did get overseas at least a little bit.
Aside from needing a replacement screw for his adjustable sight (which he got arranged), apparently this pistol worked out very well for him.
.45 Winchester Magnum is occasionally available off the shelf or online, but it's the kind of cartridge that if you want to shoot it, you really ought to handload it. To me, it would be the best caliber for one of these. To give a description of .45WinMag, imagine .45ACP, but 30% longer, with a much a much stronger case web, and loaded on average for twice the velocity, your typical 230gr projectile going at 1600fps, possibly more depending on what you like to load.
This isn't far from .44 Magnum, in fact this stated load would be slightly more powerful than the average .44 Magnum load, while being rimless, which makes it better suitable for automatics.
The Automag IV was also available in .44AMP, which can be loaded for and done today if you want, but was more troublesome in past decades, requiring proper converting of .308 Winchester or .30-06 Springfield casings. These days you can order ready made .44AMP casings from companies like Starline Brass.
10mm Magnum is a cartridge I've really not heard a lot about, but I'll hazard a guess that it's roughly comparable to .44AMP and .45WinMag, I assume you could get this kind of brass made for yourself by companies like Starline and you can use typical .40 caliber projectiles, so it shouldn't be much worse.