OK, how are you in terms of familiarity with your chording? Are you able to recognize the chords by name and have you been practicing changing from chord to chord at a regular tempo? See my Ukulele Boot Camp link below my signature.
If you are still learning chords and still have interruptions changing from chord to chord at a regular tempo, then you really can't concentrate on strumming because as the chords in the song change, you'll be interrupting your timing to make the changes.
OK, let's say you got the chords and chord changes down cold. Now it's a matter of timing. Try to figure out how many beats there are in each measure for the song you're playing. Probably 4 strums to the measure (4/4 time) or 3 strums to the measure (3/4 - waltz time). It's important to know this because your strumming, in the big picture sense, must match the timing of the song.
If you're playing "Happy Birthday to You", that's 3/4. It starts on the up beat (3rd beat "Happy...") of the intro measure and then the down beat (1st beat of the measure = ...Birthday"). You might get thrown off when it comes to the name of the person since we tend to hold out the time. But then we will go back to the 3/4 to finish out the song. If you tried to play this song with 4 beats per measure... well try it. It's not the same because you'll have to hold out "Bir-earthday" every time.
For "You Are My Sunshine" although it's in 4/4 time, the first beat is 'empty' after the first "Sunshine": 'You are my Sunshine (stop) my only Sunshine (stop) you make me happy (stop) when skies are grey...
Anyway, know the number of beats per measure, and start with simple down strums or regular up-and-down strums.
I say that working on 'performance' strumming should always come LAST after you've got the chords, chord changes, lyrics, and melody down pat with plain strumming. Once you can play the song well with plain strumming, then you can begin to try to make the song sound like what you first heard that endeared it to you.
Clear as mud? I hope not. But it does take time and practice. Remember, chords and chord changes first, then lyrics, then strumming (the way you want to perform it).
Keep uke'in',