Is the money that has reached the nisab and upon which a year has passed the only amount calculated when giving zakat, or are the monthly saved amounts added to it?

If the money reaches the nisab (minimum threshold) and a hawl (full lunar year) passes over it, zakat becomes obligatory. As for money that becomes available after the initial nisab was acquired, and a hawl has not yet passed over it, it is permissible to pay its zakat along with the zakat of the nisab that has completed its hawl, as an act of voluntary prepayment. It is also permissible to calculate an independent hawl for each separate sum.