The pattern is exactly the same throughout the song, 2 bars of C, 2 bars of Am, 1 bar of F, 1 bar of G7 and back to 2 bars of C.
Stand by Me has a really distinct rhythm and to make the song recognisable you’re really going to need to emulate that as closely as you can. If you’re new to Ukulele then I’d suggest this strumming pattern repeated throughout…
Strumming pattern (D=down U=Up)
D – DU – UDU
The as you get more comfortable thing about replacing the second down strum with a chuck/chunk/muted strum so the pattern becomes…
D – XU – UDU
Finally, if you really want to push it a little bit more, you can push it by hitting the body of your guitar for a more percussive sound.