First thing it came to me was Radiohead - How to dissapear completely and The Smiths Never had no one ever for instant depression! :)

Maybe not exactly indie in the sense you're asking for though.

sad song
 

i can feel a hot one - manchester orchestra
the light behind your eyes -mcr
the world is ugly - mcr
therapy- all time low

Be Quiet and Drive -Deftones

That song just kills my soul, if you haven't had the chance you should listen to it. It is extremely sad.

Off the top of my head I have a few random references for a variety of sad listeners. Whether you're dumped, cheated on, lost, doing hard drugs, isolated, or just seasonally moody I believe I have you covered. I'll come back if I think of anything else.
(Radiohead) Radiohead - There, There (The Boney King of Nothing)
(Early 00s Mildly Poppy Indie) Hot Rod Circuit - At Nature's Mercy
(Late 90s Sad Math Rock) Braid - Forever Got Shorter
(Folk Emo) Owen - A Bird In Hand (OR American Football - The Summer Ends)
(Relevant Shoegaze) Whirr - Formulas and Frequencies
(Surprisingly Miserable Melodic Club Music) RAC - If You Forget Me
(70s Sad Holiday) Joni Mitchell - River
(00s Mainstream Punk) Sum 41 Pieces
(Contemporary Lo-Fi Dream Pop/16 Candles on Prozac) Procession - Spring Thaw
(Late 90s drop-tuned misery) Deftones - Digital Bath
(00s indie/emo boiz) The Early November - Everything's too cold, but you're so hot

Ryan Adams - La Cienega Just Smiled... Or basically anything by Ryan Adams haha.

The Veils - Vicious Traditions

Band of Horses - The Funeral

I need some help, please.
I'm looking for a sad pop britpop alike song that I used to hear on radio between 00's to 05's. I remember that it had a piano intro very similar to Snow Patrol chasing cars or "the lightning strike" but a little bit slower and depressive but is not Coldplay. I also tried The Fray.
About Lyrics... sorry but it was 10 years ago and I didn't know that much how to "listen" lyrics in English. (Sorry about my English)
I recorded something that could help about this "piano intro" (youtube).
DdE4mKZAq78

If you want to hear sad songs, listen to Keaton Henson's 'Dear...' and 'Birthdays' albums. Brilliant, but so purely heartbreakingly emotional and oftentimes dark and sad.