Seilerbird is right about the scale terminology. Let's try it from a different viewpoint and see if you can get a better understanding of the terms he used. The scales begin on your uke at the open fret on your top string so let's begin there.
Your strings are are tuned to GCEA looking at them from the top down (which is why chords on a uke are different than chords on a guitar where the strings are EADGBE from the top to the bottom in standard tuning). So begining on your top string, each fret along that string is a note. Some notes will be sharp (#) or flat (b), what you call the note can be up to you at this point but I generally call them sharps (an example, I call the note between the G and the A, a G# although you could also call it an Ab). Remember that a sharp note will be a little bit higher than the note it follows while a flat note will be a little bit lower than the note it precedes. Let's not get to hung up on what to call flats and sharps, just use one term for now. The sharp and flat notes are the same thing as the black keys on a piano.
If the top string is G the notes on your ukelele's fretboard for that string will be like this (this range is one octave, the notes begin again at the 12th fret but in a higher octave):
0 1 2 3 4 5 6 7 8 9 10 11 12
G G# A A# B C C# D D# E F F# G
Now when Tom mentions a whole step, as between G and A, there is a G# (you could also call it an Ab) located between the G and the A. When Tom describes a half step, such as between B and C, there is no sharp or flat note to deal with. There are only two half steps, B to C and E to F.
Let's try a simple "Blues" progression called a I-IV-V, one of the most used (some may say overused) chord progressions in music.
Okay, so if this progression is in the key of G, your root note, the I note, is the G which you could play open on your top string. The next note, the IV, is the fourth note in the scale which begins with your root note of G (since you're playing in the key of G). This makes the IV note the C which is found at your 5th fret. Note that we did not count the sharp and flat notes to go from the I to the IV, do not confuse counting frets with counting up your scale.
In the I-IV-V progression the next note will be the I note again, so you would come back down to the G. The progression then moves to the V note which is the D at the 7th fret before returning to the G, the I note.
When I began playing I was 12 and it was easier for me to remember the frets on the guitar than the scale progressions so a I-IV-V to me became a O-5-7 which was my way of counting frets. I did this for years until I learned music theory.
I hope this helps.