There are several common methods to do it. To indicate fret positions I use two digits: string/fret: 14 = 1st string, fourth fret. Above the 9th fret, I use capital letters: A=10th fret, B=12th, etc. Sometimes I use a third digit to indicate the finger number.
Classical guitar method: play in positions. In 2nd position, play across the strings until you reach high D (15), then shift into 7th position for the next three notes on the 1st string, then shift into 12th position to play the last three notes. Coming down, you probably want to shift back down to 7th position on the 1st string (a more comfortable area to play than 12th), but then you can play across the strings until you need middle F, when you shift down to 2nd position again:
Ascending: 30 32 34 35 23 25 12 13 15 (shift ^) 17 18 1A (shift ^) 1C 1E 1F
Descending: 1F 1E 1C (shift v) 1A 18 17 2A 28 27 39 37 (shift v) 35 34 32 30
or just reverse the ascending sequence. There are no guide fingers for the descending shifts.
Going up, at 12 keep your index on the 1st string, to use as a guide slide to 17; similarly, keep it on 17 until the slide to 1C. These are all silent slides: the guide finger doesn't press the string down to the fret, and the slides should be very quick to minimize the break in legato.
The point of playing across the strings in the middle of the neck in the descending pattern is that playing scales is usually much different from playing melodies: when playing melodies, you want the hand to stay as much as possible in the same positions, with all the notes needed (for the nonce) right under the fingers.
Guitar staggered method with sliding index:
30 321 (slide) 341 352 373 251 (slide) 271 282 2A4 171 (slide) 181 1A2 1C3 1E4 (slide) 1F4
or …282 2A3 2C4 181 (slide) 1A1 1C2 1E3 1F4
Where you decide to stagger and slide is up to you; the main point is that climbing the neck this way uses incremental shifts, mostly with index slides or when crossing strings. Since a uke has only four strings, and on reentrant ukes, the 4th string pitches overlap with the 1st and 2nd string pitches, to cover a span of two octaves so quickly, you have to make aggressive use of incremental shift opportunities. On guitar, or even linear uke, you could play only three notes per string, and you wouldn't have to index slide on each string.
Campanella method (Petey can describe this better than I, since I rarely play scales this way):
30 322 20 211 40 10 (shift to 5th pos.) 273 451 152 (slide-shift to 10th pos.) 2C3 4A1 1A2 1C3 1E4 1F4
or
30 322 20 211 40 (shift to 7th pos.) 393 271 282 471 172 (slide shift to 12th pos.) 2D3 4C1 1C2 1E3 1F4
For the three notes prior to the slide-shifts, keep the fingers on the strings when planted (and even keep them depressed, if you want the corresponding notes to keep ringing); then release pressure and slide the entire hand shape up the neck, maintaining contact with the strings. It's really nice when you can either shift while playing an open string, as in the first shifts, or keep the same hand shape in successive positions, as in these slide-shifts, but it doesn't always happen.