Here's what I would do...
I'd get a 45-70 case and a loaded round. And (for that matter) a 348 Win., and maybe even a 375 Win, or just about any other rimmed cartridge that is similar in size to the 8x56R. I'd then compare these rounds/cases with the 8x56R.
I think that the first thing I'd try to find a match for would be the case head diameter, and then I'd go for rim thickness. If I got really close there, then the next thing I'd do would be to check out the loading manuals to see what kind of pressure these different cartridges produced.
If the new rounds were equal to, or lower than the 8x56, I would start to monkey around with the clips, seeing if I could load the different rounds into the clip, and then see if the loaded clips would fit into the rifle. If they don't fit, but are close, you can probably work with the action to make it happen.
I realize that there are feed rails that would probably have to get modified and altered for the different ammo, but we're just looking at the feasibility here, right?
I would tend to think that the 45-70 if loaded normally, and not to the levels which a Ruger No.1 could handle - would be a decent round to shoot for. If the rims were similar, and if you could manage to get the rounds to load in the clip, I'd say that you probably have something that you could probably modify into a shooter. The same would go for a 348, or a 444, etc.
Rebarrelling would be expensive, though. The barrel need to have the shank cut to the proper dimentions, threaded to the proper pitch, and special breeching duplicated and perhaps you'd even have to try to match the contour of the old barrel if you wanted to use the old stock/furniture.
The guy doing the machine work would also probably have to work on the feed rails or other internal parts of the action, so this would add to the cost, and then again, we have to think about some sort of sights to go along with this deal.
But, given deep pockets and the proper materials, I would bet that this really wouldn't be such a difficult project. Again, my concerns would center on pressure and matching the case rims as closely as possible. It's easier to machine the bolt face bigger rather than the other way around!
Perhaps this rifle could be rebarreled to the 7.62x54...? Plenty of that kind of ammo around, you'd just have to check the compatibility of the two rounds.
Worth looking into, eh?
Good luck!
Unkel Gilbey