Bismihi ta'ala
Firstly, we should refrain from using the word dowry, because in English it is a payment given by the bride to her husband.
Whereas, in Islam, it is from the groom to the bride, and not a payment, but a token and sign of loyalty and truthfulness, which is why in the Quran and Sunnah it is called "Sidāq".
We also call it Mahr, and in English, we can call it bridal-gift or nuptial-gift.
Once the marriage contract (nikah / 'aqd shar'i) is done, it becomes the husband's duty to fulfill the contract and what he comitted to. This obligation depends on the agreement the bride and groom had. Was there a clause in the contract, like dividing the mahr into "muqaddam" or immediate, and "mu`akhar", which is the deferred amount.
Is the husband capable of giving it immediately. Has the wife demanded it? Can the husband give it in installments. These are different scenarios that could occur, so it would depend on the circumstance.
Some cultures also have it to be given on the day of the nikah as well.
So, from a shar'i perspective, it becomes wajib upon the husband to give the mahr that he agreed on, whenever it is demanded by the wife, whenever he is able to, and according to whatever agreement they had.
And Allah knows best
