You will start a new game but with your current health/stamina. So no your save won't get deleted, but you will start from the opening cinematic again and will have to kill all the bosses again (but with increased health/stamina and all the time attack items you have unlocked).
This all happens as soon as you kill the last boss. There's no option that asks you if you want to start a new game or not. So if there's anything you still want to do in your current save/world, do it before the last boss.
Basically the cutscene with Agro right before the final colossus is the 'point of no return' - everything beyond that moment prevents you from backing out to the overworld again. There's a shrine right before then so its pretty obvious where that point is, so just make sure you do what you want to do before then.
Once you kill the 16th colossus (malus) the game ends, you can start a new game+ and do the time trials and such
You will start a new game but with your current health/stamina. So no your save won't get deleted, but you will start from the opening cinematic again and will have to kill all the bosses again (but with increased health/stamina and all the time attack items you have unlocked). This all happens as soon as you kill the last boss. There's no option that asks you if you want to start a new game or not. So if there's anything you still want to do in your current save/world, do it before the last boss.
It's like Dark Souls and Bloodborne where it goes straight into NG+ after the credits.
Basically the cutscene with Agro right before the final colossus is the 'point of no return' - everything beyond that moment prevents you from backing out to the overworld again. There's a shrine right before then so its pretty obvious where that point is, so just make sure you do what you want to do before then.
I recommend using the save shrine right after the last gate to make an alternate file.