### Step-by-Step Solution

#### Part (1): Find the selling price per bike in August.

Let's denote the selling price per bike in August as $x$ yuan. The information given can be translated into the equation based on the sales amount and the increase in price:

- The equation is derived from the fact that the total sales amount in July ($22,500$ yuan) at a price of $x-100$ yuan per bike equals the total sales amount in August ($25,000$ yuan) at a price of $x$ yuan per bike, assuming the quantity sold remains constant.

Thus, we have:
$$\frac{22,500}{x-100} = \frac{25,000}{x}$$

Solving this equation for $x$, we proceed as follows:
$$22,500x = 25,000(x - 100)$$
$$22,500x = 25,000x - 2,500,000$$
$$2,500,000 = 2,500x$$
$$x = 1,000$$

After solving, we find that $x = 1000$ yuan is indeed a solution that satisfies the original equation.

Therefore, the selling price per bike in August is $\boxed{1000 \text{ yuan}}$.

#### Part (2): Find the cost price of each mountain bike.

Let's denote the cost price of each mountain bike as $y$ yuan. According to the problem, the selling price in September is $15\%$ less than in August, and despite this reduction, the store still makes a profit of $25\%$ on the cost price.

The selling price in September can be calculated as $1000 \times (1 - 15\%) = 850$ yuan.

The profit made on each bike sold in September is given by the difference between the selling price in September and the cost price $y$, and this profit is $25\%$ of the cost price. This gives us the equation:
$$\frac{850 - y}{y} \times 100\% = 25\%$$

Solving this equation for $y$, we proceed as follows:
$$\frac{850 - y}{y} = \frac{25}{100}$$
$$850 - y = \frac{1}{4}y$$
$$850 = \frac{5}{4}y$$
$$y = 680$$

After solving, we find that $y = 680$ yuan is indeed a solution that satisfies the original equation.

Therefore, the cost price of each mountain bike is $\boxed{680 \text{ yuan}}$.