Before talking about playing by ear. One needs to understand what is commonly called ear training. As the WikipediA definition for Ear Training mentioned, ear training has an active and passive roll.

PASSIVE EAR TRAINING is the ability to recognize and name intervals, chords, chord progressions, melodic sequences, rhythm, etc... An example of passive ear training would be if someone played the first two notes of the song "Here Comes the Bride" you can recognize them as an interval of a perfect fourth. You might not know the actual pitches, that would be perfect pitch, the ability to identify or re-create a given musical note without the benefit of an external reference.

The hands are the weakest link in the Ear to Hands to Brain combination that is used to play music (by ear) and the hands need to be trained to follow the ear. It is the hands training that takes the longest.

The ear can remember stuff after ONE time hearing it. The Brain two or three and the hands to develop the muscle memory take about a few hundred times. How else can we recognize melodies after hear it once?

When practicing you should be doing stuff to train the hands to follow the ear. That is the sequence for developing this connection:

The Brain tells the Hands what to play and the Ear listens. DON'T practice by ear. Have a plan and organization of the material you would like to cover.

Practicing your scales in intervals of 2nds, 3rds, 4ths, 5ths, 6ths, 7ths, etc... Some more common than other for melodies. Practice both ascending and descending. I would go with 2nds, 3rds, and 6ths first.

When performing or "Playing by Ear" the sequence is:

Ear tells the Hands what play and you can involve the Brain if needed or wanted after the fact.

The hands will not do anything they have not done before.

So to be able to play by ear you either:

1) Hunt and Peck enough to train your fingers to follow the ear – or –

2) Have a plan (brain) to train ear and hands to work together, thus making the hands to the ear connection quicker. Creating a plan is where a teacher can really help. And, if you've had bad experiences with teachers don't give up on them just try another, good teachers are out there. Typically jazz musicians will be the best teachers in the playing by ear department. Remember playing by ear does not mean practicing by ear. Practicing and learning are different then performing.

Here is an example. The first two notes of "Here Comes the Bride" are a perfect fourth. Try putting any finger on any note of the ukulele and then play the second, "Here Comes..." correctly the first time. If you can't the fingers haven't been trained enough to "Play by Ear", to obey the hands for that particular interval/sound. This is then where the brain comes in and intellectually knows where the interval of a perfect fourth is and trains the hands to follow the ear.

Same goes for chords, chord progressions, melodies, rhythms. The process starts with the brain telling hands what to do and the ear taking it all in.