The only functional difference in the action is the extractor, which is available thru many sources around 10 dollars. OEM extractors can be modified for specific uses, such as in rimfire conversion. Centerfire extractors can be more straight forward, use as is or modify slightly for the case of the day.
I have two previously in 375 H&H No.1's, one is a 257 Roberts and the other is a 22LR. The forearm that you have will be for a fairly large barrel(375), so that enters somewhat into a build, or to even use reuse it. Stocks are stocks for the most part. I have noted over the years, no.1 builds are as varied as the wind.
Some will buy a new extractor then hunt for a take off no.1 factory barrel. This still has some work to do with fitment of the new to you oem barrel to your action. Some will, some won't; time up so that the barrel is tight; headspaces, gapspaces, and the sights are straight up. So guess, you pays your money and takes your chances with a takeoff barrel. Lots of folks enjoy the factory look.
If you want a new barrel, lots of nice barrel blank makers around. The No.1 barrel blank has to have the usual things done by the smith, threading, crowning, and chambering...........and finish of your choice. The No.1 barrel has a few things that the typical bolt gun does not, the extractor has a recess in the barrel face and the rib/scope mounts are affixed to the barrel(4 screw and the optional 2 studs), the cost of a Ruger No.1 rebarrel with new is about doing a pre64 Winchester 70 in labor in my opinion. Of course, the rib can manage a folding rear sight easily, but the front banded oem ramp sight is more labor than some to put on.
Certain aftermarket tapers mimic the factory tapers very closely, but of course, most blank makers offer a lot of contours free or for a minimum fee.
The real beauty, other than its cosmetics, is its versatility, the action can serve its purpose with a lot of chamberings. The forend seems to be the dividing factor with some, several versions to choose from there. Lots of folks like a full custom no.1, but the costs(unless you can do the work yourself) can run into $, a simple reworked No.1 with factory wood is indeed a very nice rifle.
Best of luck with your procurement and mods.
FWIW,
Roof