I think the advice given of "hope for the best, plan for the worst" is the best. Carry the uke on your shoulder, be polite, and should you get hassled, cooperate. If you're prepared, you'll have your carry-on ready to gate-check for the "worst".
Leaving Honolulu on United, my son and I were going to carry our 2 new ukes on in addition to the carry-on and personal item, which normally isn't a problem. But they announced pre-boarding that the plane was completely full, and there may not be enough room for all carry-on luggage, so the absolute limit was one and one, and if you could help out by gate checking your carry-on, they'd appreciate it. I resisted, but after about 4 of those announcements, I figured they meant business, and we chickened out and gate checked our carry-ons (I never check luggage, but didn't want to get to the jetway entrance and hold up the line).
Both ukes, a concert and soprano in gig bags, fit in the gap between the seat and fuselage, so I didn't even need to put them in the overhead; and, gate-checking the bags turned out to be fine, they were there when we got out the door, and we walked right by all the people waiting for checked baggage, as we always do...