I'm guessing this progression is F - F6 - Fma7 - F6 in the verse? When I play this I just add my pinky on the 4th fret, 3rd string without moving from the F6. If you're new to the ukulele this may take a bit of practice but you'll get it eventually
ETA: Whoops, I read that as Fma7 not Fmi7 because that's the version I play (transcribed from Joe Brown in Concert for George). If your version of I'll See You in My Dreams is different, just disregard everything I said
Fm7 is 1313. To me the easiest way to play it is barre your index finger across the first fret and then use either your middle and 4th fingers (or 4th and pinky) to fret the notes on the C string 3rd fret, and at the A string 3rd fret...whichever is easiest.
Further up the neck, you'll find some more voicings at 5546 (a Dm7 shape played 3 frets up) and at 8888 (a simple barre of the index finger over the four strings on the eigth fret).
For the minor 7 chords with the 1313 shape of the Fm7, I use the thumb to play the G string, middle finger on the C, index finger on the E, and ring finger on the A.