If I understand what you are saying, it is incorrect. All the fretted instruments that have been mentioned (as well as piano and other keyboard instruments) play enharmonic notes—notes with different names that sound the same—using the same fret and string positions. In other words, Bb/A# are the same note on the ukulele and the other instruments mentioned.
On string 1 (A), the names of the notes on the various frets are:
Fret
1 = A#/Bb
2 = B
3 = C
4 = C#/Db
5 = D
etc.
It might be useful to have you ukulele to play as you read.
Only in certain situations is a # or a b related to ascending or descending. It is best for starters to understand that every note on every string and fret is equal and independent to every other one. The reason that a certain noted might be called A# one time and Bb the next is farther down the road of music theory and relates to music notation conventions decided hundreds of years ago. A# has nothing inherently to do with A, it is just another note, a note that could have had a less confusing name.
As Alan mentioned earlier and Mfturner just mentioned, the relative distance (number of frets) between one note and another is what is important, not the note names. For example, if we want to play the first three notes of "Three Blind Mice" (which are the same as the three notes of a major scale, do re mi backwards) on the first string we play the note on the fourth fret, then the note on the second fret, then the open string (the 0 fret). It doesn't matter what the names of the notes are, it is the relationship of the notes/frets—play the 4th fret, skip a fret, play the next fret (2nd) then skip a fret, play the next fret (which in this case would be the 0 fret).
You can play the same melody starting on any note on any string as long as you start on the fourth fret or higher. If you start on the 9th fret, you use the same relationship of notes and play the 9th, skip a fret, play the next fret (7th) then skip a fret, play the next fret (5th). Start anywhere (above the fourth fret) on any string and try the pattern. You will always get Three Blind Mice, and it has nothing to do with the names of the notes. (Knowing the names of the notes is useful, but not important for what we are talking about here.)
Unless a person has perfect pitch (the ability to name a note just by listening), no one can tell if the names of the notes you are playing contain sharps or flats. But everyone will know that you are playing Three Blind Mice.