[RPG] Does a bard know when a character uses their Bardic Inspiration

No, Bardic Inspiration can't cause a critical hit

Cutting Words states, "...subtract the number rolled from the creature's roll.", so the number is being applied to actually change what you rolled on the d20.

Bardic Inspiration states, "add the number rolled to one ability check, attack roll, or saving throw", so the number is just another bonus on the attack roll.

The difference in phrasing is very small but important here. Terms like, "attack roll", "saving throw", "ability check", are common phrases in 5e used throughout the book. PHB, p.195, explains that you make an attack roll by rolling a d20 and adding appropriate modifiers. If it weren't for the Jeremy Crawford tweet it would be sensible to assume that the, "roll", in Cutting Words refers to the, "attack roll", but that tweet exists so instead it refers to the actual roll of the d20. Bardic Inspiration specifically says, "attack roll", so we can be relatively certain that it is just another modifier added to the roll.

The spell Bless uses the same phrasing as Bardic Inspiration, "...and add the number rolled to the attack roll or saving throw", and there is no reason to believe that it grants (or bizarrely prevents) critical hits.

According to Jeremy Crawford in this conversation...

If a player character ability relies on knowing a monster's roll, a DM may say what the roll is or show it.

When asked to confirm that "[the] DM must communicate only the number on the dice (without [any added] modifiers)," Crawford responded...

That's correct.

So the bard (or any creature using a similar feature) is told or shown the number rolled on the die, not any modifiers which would apply to the roll. To be clear, then, they are not told the total.