It's made after 1940, when the soundhole decoration was changed from black-natural rope binding to white-black-white rings. The (wonderful!) 13rd fret also says loud that this on is made after the early 1940s. The body's probably also slightly shallower than earlier rope-bound ones.
White nuts and saddles, and body binding were usually reserved for uke-2 and uke-3.
Gibson was very sketchy about it's different styles, the uke-3 in fact toning down so much it became almost indistinguable from a uke-2 by 1938. But in fact, by 1940 Gibson dropped the uke-2 name, and called it uke-3 until around 1954. In spite of this, we usually call these instruments without fretboard binding and back binding and with smaller fretboard dots uke-2s.
Headstock logos are VERY confusing when it comes to Gibson, in spite of what many websites claim. Yes, this is the 1930s white 'script' logo, but it was used for a very wide period of time, well into the 1940s.
If you look inside with a flash light, there might be a code stamped or written in pencil on the neck block, which might help dating it. It's not a serial number, but a Factory Order Number meant to trace production batch and cost (so numbers were assigned haphazardly and often reused) but it can give a good idea. Uke-1s usually don't have them, TU's almost always have them.
