Two things that are probably key (no pun intended) to being comfortable playing Bb on a uke. First, make sure the uke is set up. If the nut is high you have two problems, it's hard to finger a barre across the first fret (more on this in a second) and it probably won't sound right to your ear even when you do (because some of the notes are being pulled sharp by the high nut).
Okay, assuming our uke is set up properly, why barre at the first fret when we're fingering the third and fourth strings farther up the neck anyway? Let's say we're playing in the key of F, where your primary chords are going to be F, Bb, and C. If you finger the open F chord with your index and second finger and then try to pull your index finger back (or tuck it down) to cover the first two strings at the first fret, it changes the angle of your hand and throws your second and third fingers out of position. Sure, you can
do it that way, but your whole hand is changing angle with respect to the fretboard. If, instead, you simply lay your index finger down flat across all four strings at the first fret, the angle of your hand and wrist doesn't change, you move your second finger over one string and plop the finger down on the third fret, fourth string. Reverse the process to go Bb to F. It's a very natural transition after you do it a few times and you'll notice that your hand doesn't change position at all. It stays pretty much perpendicular to the strings, arched about the same amount, etc.
You can throw in the C from either of the above chords by simply raising all fingers except the third or the pinky, either of which you drop down on the first string, third fret, again, without changing position of the hand and wrist very much. (Similar story for the C7, just curl the index back to put the tip on the first string.)
On any stringed instrument, the less that you have to change the position, shape, or angle of your hand and wrist, the faster you will be able to play with less fatigue. (Actually, I suppose that's probably true of any instrument, not just stringed instruments.)
So, if you're having trouble with the Bb / Bb7 (B, B7, etc) and you are trying to fret just the first two strings with the tip of your index finger - give the barre a try. It may feel wierd at first if you've been doing it the other way for a long time but it will probably become your favorite Bb fingering pretty quickly. Or not...I forgot to wear my fortune-teller's cap today.
HTH,
John
Any time you can make a transition by moving only fingers