There is no specific time for paying zakat. However, paying zakat in Ramadan is highly encouraged because it is the most blessed month in the year. You need to pay zakat on the money you possessed for one year, provided that the money crossed the threshold amount.
Paying zakat is so important that the Holy Quran speaks about it in many places. Everytime Allah ordered us to pray salat, it is also mentioned to pay zakat. The Hadith also states in many places to pay zakat e.g.
teach them that Allah has made it obligatory for them to pay the Zakat from their property and it is to be taken from the wealthy among them and given to the poor.
-Bukhari Book 24 Hadith 1
The punishment for not paying zakat is severe. The Quran states that:
And let not those who covetously withhold of that which Allah has bestowed on them of His Bounty (wealth) think that it is good for them (and so they do not pay the obligatory Zakaah). Nay, it will be worse for them; the things which they covetously withheld, shall be tied to their necks like a collar on the Day of Resurrection. And to Allah belongs the heritage of the heavens and the earth; and Allah is Well‑Acquainted with all that you do”
[Aal ‘Imraan 3:180]
“And those who hoard up gold and silver (Al‑Kanz: the money, the Zakaah of which has not been paid) and spend them not in the way of Allaah, announce unto them a painful torment.
The Hadith says the following (out of many) about the punishment of not paying zakat:
Abu Dharr reported:
I went to the Messenger of Allah (ﷺ) and he was sitting under the shade of the Ka'ba. As he saw me he said: By the Lord of the Ka'ba, they are the losers. I came there till I sat and I could not stay (longer) and (then) stood up. I said: Messenger of Allah, let my father, be ransom for you, who are they (the losers)? He said: They are those having a huge amount of wealth except so and so and (those who spend their wealth generously on them whom they find in front of them, behind them and on their right side and on their left side) and they are a few. And no owner of camels, or cattle or goat and sheep, who does not pay Zakat (would be spared punishment) but these (camels, cattle, goats and sheep) would come on the Day of Resurrection wearing more flesh and would gore him with their horns and trample them with their hooves. And when the last one would pass away, the first one would return (to trample him) till judgment would be pronounced among people.
Muslim Book 12 Hadith 37
As for reaching the due date, you should repent to Allah (SWT) for not paying zakat in time and then pay the zakat. Allah is the most Merciful, the most Forgiving.
“And He it is Who accepts repentance from His slaves, and forgives sins”
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said that:
Zakat is an act of worship towards Allah and it is the right of the poor. If a person withholds it, then he is violating two rights: the right of Allah and the right of the poor and others who are entitled to zakaah. If he repents after passing the due date then he is absolved of the obligation to fulfil the right of Allah