record somone singing the song correctly, and then she can sing along to the recording in the privacy of her own home without having to worry about anyone listening in judgment.

Also, if it's the same problem each time – i.e. coming in on the last beat of the bar – then maybe your friend could try just clapping or stomping a 1-2-3-4 count, coming in with whatever word or syllable the line starts with (but no more of it) and immediately repeating over and over and over and over again.

Singing teachers, I guess, must deal with this sort of thing all the time and have exercises to improve timing and coordination - maybe she should book a lesson or two.