I have helped Gullivarrr 2 times already, but never got any reward in the mail the next day.
However, Gulliver does send me rewards in the mail.
Could this be a bug or purely my own fault?
That's pretty strange. I've encountered Gullivarr twice now, and both of the times I got something in my mailbox. I have the feeling sometimes it takes more than one day, but maybe that's me.
If you gave him his phone and watched him call his crew/thank you, you should be getting your reward. Unless you somehow force quit the game before the autosave came in. Or time traveled backwards? I don't TT, but I hear that TTing causes the flowers to not be watered so maybe it could be this?