Zakat is due immediately on your Zakat anniversary, whenever that may be. (Your Zakat year should begin on the date your wealth equals or surpasses the Nisab. If your wealth equals or surpasses the Nisab after one lunar year has passed (also known as Hawl), then Zakat will be payable. In this instance, you will use this date again for the following year). Paying Zakat should not be delayed unnecessarily.
Regarding Ramadan:
Paying Zakat in Ramadan is not necessary, although giving charity in this month guarantees greater rewards. If you wish to pay Zakat in Ramadan and your actual Zakat anniversary comes later, you can do so on the condition that when your actual anniversary arrives, you calculate whether you have paid the right amount and make up the deficit if need be. This is based on the principle that Zakat can be paid in advance – discussed here.