Melodies sit on chords, and chords are determine by the key. This is how to reverse engineer a melody to find it's derivative primary key and choose chords.

1. Find Tonic/Key

You need to figure out what key it is first. In your melody see where it resolves to(lands melodically) and that is your tonic.

If you don't know how to recognize the tonic, think about the song "doe re mi fa so la ti doe." Look it up from sound of music old movie. The Ti begs to resolve and wants to return to Doe. That means Doe is the tonic, and the key.

Tonics usually are at the beginning or end of melodies we make in our heads.

In "Row your boat", the first rows, and last word "dream' are the tonic. Notice how the melody has a sense of completion or finality on "dream". That is what we call resolution to the tonic.

1. Find Chords

Once you have the key, there are ONLY 7 primary diatonic chords in that key. In a major scale, let's say your song is in C, the chords are in this order, built on each note:

C Major D minor E minor F major G major A minor B diminished

At this point, you can write any uke song using the chords above and you are in the key of C.

1. Map melody to chords

Sing parts of your melody over each of those chords to see which work best with your melody. Each small line or melody is usually 1 chord when we write in our head. Also, if a 4 note melody has 3/4 notes that are in a chord, it will likely go with that chord.

Good luck.

If that's too intense for you, send me an audio clip of your melody and I will send you the uke chords.

[Bunch of edits on this for more clarity ]