Yes, the tuner should be able to sense the note name and tune just fine, in either octave. Some tuners with a more sophisticated display will tell you the octave (low-g is G3, and high-G is G4) and also the audio frequency, if memory serves, low-g is like 197 hz and high-G is like 394 hz there abouts, similarly with the concert-pitch 'A' which is 440 hz, which is A4 and an octave below, A3 is 220 hz. with the octave above A4, which is A5 is 880 hz.
These are all 'harmonic' frequencies, and as such the tuner should be able to understand them unless the batteries are weak.
Just in case if you are not sure what is on your uke, tune it up and then fret your C string (3rd up from bottom) at the 7th fret, if that fretted note is the same pitch and octave as your G string (4th up from bottom), then you are in high-G tuning (it will be what is called a 'unison'), but if the notes are similar but not quite the same, then you probably have a low-G string, and are in low-g tuning.
Also if your G string is 'wound' like the fatter strings on a guitar, then, you have a low-g.
If you are still not sure, and have an iPad or iPhone, there is a free instrument tuner app that can help, if you use the info above about octaves and frequencies, it's called 'insTuner' and it can use the built in mic to listen to your instrument so you can tune it.
-Booli