The melody for the verses of "Heaven Only Knows" is something like:
(I used the first verse as a reference, but I'm missing some nuances.)
And the first half of the melody for the verses of "You Won't See Me" is something like:
(I guessed on the keys for both.)
The melodies are fairly similar, but the more recognizable resemblance is one of rhythm. In simple terms, both songs have three syllables, some rests, two syllables, some rests, and then either a second line or the rest of the first line:
I don't know if Jeff Lynne intended the melody of "Heaven Only Knows" to be similar to that of "You Won't See Me," but there are a number of other Beatle references in the ELO catalogue, and the two melodies certainly are alike. It seems to be at least a subconscious borrowing.
I also noticed that "change" in the line "But I think that it's about to change" is sung with a glissando (E to D, I think), which musically represents that shift.