Learn Italian: Se non te ne accorgi tempestivamente, il danno è grave. - If you do not notice it promptly, the damage is severe.

It replaces the thing being noticed. The base verb is accorgersi (to realize or notice). Because you always notice 'of' something in Italian (accorgersi di qualcosa), the particle ne is used to replace di + the topic. So, te ne accorgi translates to 'you notice it'.

Why did the reflexive pronoun ti change to te here?

In Italian, when indirect object or reflexive pronouns (like mi, ti, ci, vi) sit right before lo, la, li, le, or ne, their ending vowel changes to an e. Therefore, ti + ne combines into te ne.

Is the adverb tempestivamente related to the word for a storm?

Yes, it shares a root with tempesta (storm), but today it simply means 'promptly' or 'in a timely manner.' It is an excellent B2 vocabulary word used to describe taking action at the exact right moment to head off a crisis or 'storm.'

Could I use the future tense for the second half of the sentence, like 'il danno sarà grave'?

Yes! This is a 'first conditional' sentence expressing a very real possibility. You can use the present tense in both halves to state an absolute fact, or use the present tense in the se clause and the future tense in the main clause (il danno sarà grave) to emphasize a future consequence. Both are completely correct.