I don't want to get into a gut-thumping contest here (I believe Saxon Pig would win that one

) but
my gut says it's the original nickel finish. The hammer and trigger do not look nickeled to me (a dead giveaway of a refinish), although the trigger seems to have lost most of its case color. The flecks on the sideplate and slight wear at the sides of the muzzle look consistent with an original gun of this age to me, particularly considering that it was owned by someone's great-grandmother. It definitely doesn't appear "in the white" as if the finish has been completely removed.
The pearl grips are unlikely factory since they have no medallions but are very nice and appear unchipped. I have a similar nickel and pearl .38 (the middle gun in the picture below) which I bought via Gunbroker in 2011 for under $400, but mine is of a later vintage.
http://smith-wessonforum.com/s-w-hand-ejectors-1896-1961/226247-amazing-colossal-ladysmith.html
Your wife's revolver would probably shine up nicely with a light application of Flitz (a mild polish paste) or you can leave it as-is, with its history intact. Either way I'm happy to read that you intend to keep it in the family - try to find out more about great-grandma and pass on the info to your heirs when the time comes.
*I see David Wilson replied while I was writing. I had noticed the slight line at the front of the sideplate but the pictures aren't quite detailed enough for me to decide it the edge has been rounded, or if it's just not quite snugged down flush. I'm still thinking it's original nickel.
As for shooting it, stick with standard velocity .38s and you'll be fine. Target wadcutters are nice and mild and accurate and fun to shoot. +P ammo probably wouldn't hurt it but why chance it? And, I think it would be more pleasing to great-grandma's memory to use the same kind of ammo she might have.
**Also, is it fair to say that it is
not insulting to call you Pig?
