Does zakat become due for the first year only, or throughout the entire period of holding the money, after taking a sum of money from a brother to expand a business and not giving him profits in the first year?

If the money was a loan, your brother is not entitled to any profit; he is only entitled to the principal amount loaned. Any stipulated increase or one that occurs during the loan period is considered usury (riba) and must be deducted from the loan amount. However, if the contract was a Mudarabah (profit-sharing) agreement, then your brother must pay Zakat on his share of the money, and he should pay it from the profit before it is divided. You must pay Zakat on all your trade goods and money that have completed a full year (hawl) if they reach the minimum threshold (nisab).