When you learn to sing as a child you don’t have music in front of you. You know the note you want to sing and you quickly learn how to produce that note. Generally you’re not even aware of the key you’re singing in and you can easily sing the same song in any key within your vocal range.
When you learn an instrument, you normally start off by relating the note you play on the instrument to a dot on a stave. And unlike singing, each note has a distinct finger/valve/key position, so you are very much more aware of key and pitch. And there are certainly easy keys and difficult keys, although these vary a lot by instrument. However, one of the other things you do as a classical musician is to learn your scales and arpeggios and these are a great starting point for learning to play by ear.
If you can play a scale, you can play The first nowell because it consists almost entirely of scales up and down, apart from four jumps which are always between the same two notes.
First pick a key that’s easy on your instrument, say C. The first thing you need to know is the starting note; if you hum through the tune relating it to the standard C major scale, you’ll quickly work out that it starts on the 3rd note of the scale, E. From there it’s one note up or one note down each time, until you get to fields where they lay, where you jump from C back down to G, 4 notes below.
Play this through a couple of times until you get used to it. Then change to a different key, perhaps G or F. The goal is not to learn to play The first nowell but to learn to listen and relate what you hear in your head to what you play with your fingers.
Next up, try Praise to the Lord the almighty the king of creation. It has a few more jumps, but still quite a lot of consecutive notes. In fact consecutive notes are more common than you might expect and most melody lines don’t jump around randomly but rise and fall gently in pitch.
Having started with some fairly straightforward melodies, pick another one and do the same thing. The key, as always in music, is practice. The more you do it, the better you become and the less you have to think about doing it.
The next stage is learning to harmonise a melody.