Well, no luck. I've contacted the radio station it played on, spent hours on lyrics websites, sampled hundreds of artists. I'm starting to think this song will only reappear once I have stopped wanting to find it. I almost don't even think it exists at all.
If it helps at all-
me and my dad used to listen to the ambient show 'Hearts of Space'. I don't think I recorded it from this show but I could be wrong.

Do u have any recording of it even a small one??

Can you hum a bit of it? Or, transcribe a portion of it into ABC Notation? I'd be interested in helping, given some better clues. I have a sibling whose a huge dreampop fan, as well.

What on earth is that, "dream pop"? To me it's not very clear how this "sub-genre" is defined.

Is this something like radio-friendly pop, e. g. such as "Breathless" by the Corrs?

More like Cocteau Twins or Ice Cube, with a much more modern example, The Birthday Massacre. Very atmospheric, slow, etheric pop, a decedent of Dark Wave and Etheric Wave.

Cocteau Twins? Well, now I see clearer, thanks.

Well I was originally going to try and re-create it, as I make music myself and have recording equipment, but my laptop was destroyed (by me, don't ask) and I can't at the moment. I am not adept at written music but here are the notes to the chorus, if you start on C:

CC Eflat C G(hold)
(?) and John and (?)
BflatBflat Dflat Bflat F(hold)
(?) and Johnny (?)
CC Eflat C GA(holdA)
You and John and Jeannie(?)
Bflat C FG(hold)
Ride in my carrrrr

SOLVED
http://www.youtube.com/watch?v=VCW3rAPov_4

I was way off on the lyrics. I apologize.
Almost cried when I heard it. Funny thing is, even though I haven't heard it in ten years, it sounded exactly the same as it did in my head!

Thank you to everyone for their efforts.

Just to be clear (sometimes youtube links stop working), the song you were looking for was "You and John are Birds" by Piano Magic