Hi Sausagefingers.
I'll take a stab at identifying Grandma's Nugentator.
I think it is a Gibson L-48 which was their budget model archtop marketed to students. Right now I'm thinking it is more likely a mid 50's model than a 40's model. The L-48 was produced by Gibson from 1948 til at least 1969 according to my shipping totals literature and I certainly haven't sorted through all the Gibson literature I have from this era. However, I have taken a look at a late 1940's catalog, a mid 50's catalog and an early 1960's catalog and this is what I see.
Let's start with the late 1940's:
 
Notice that this model has a more rectangular tailpiece than yours and appears to have a two ply pickguard.
Now for the 1956 model:
 
I think we have a match in terms of the pickguard and the tailpiece on this one.
If you take a look at the early 1960's model, you'll see that they have returned to the more rectangular tailpiece:
 
So, without doing exhaustive research on your guitar, my best guess is that it is a mid 1950's L-48.