You can't take damage until you know how much damage you're taking
Here's how the taking damage is described in the rules:
Whenever a creature takes damage, that damage is subtracted from its hit points.
You can't subtract a number until you know what that number is. Therefore, at the time damage is dealt, you (the player) know the damage amount. This is reflected in the way damage is typically announced in actual play: the DM just says "you take 10 slashing damage", and after that sentence is your opportunity to use Projected Ward. They don't say "you're about to take slashing damage; anyone have a response before I announce the amount?" It wouldn't be impossible to play that way, but I've never heard of anyone who does it.
Of course, your character has no concept of hit points, so strictly speaking, your character doesn't have precise knowledge of the amount of damage being dealt. However, it is entirely plausible that your character can tell the difference between a glancing blow that deals 5 damage vs. a solid hit that deals 15, even when watching someone else get hit. (Think about all the times you've seen videos of something unfortunate and cringed reflexively even before the unfortunate thing actually happens.) In that sense, I wouldn't generally consider it metagaming if you make decisions based on your out-of-character knowledge of damage amounts, because this knowledge is plausibly reflected by in-character observations.
