To minimize the total time, we can multitask during the water boiling process. Here's a step-by-step plan:

1. Wash the kettle and fill it with cold water, which takes 2 minutes ($2$ minutes).
2. While the water is boiling, which takes $15$ minutes, we can:
- Wash the teapot and cups ($2$ minutes)
- Get the tea leaves ($1$ minute)
- Given the boiling takes $15$ minutes, we can do these tasks simultaneously without needing extra time.
3. Brew the tea, which takes 1 minute after the water has boiled ($1$ minute).

Adding the durations of these steps, we have $2 + 15 + 1 = 18$ minutes.

Consequently, the shortest operation time is $\boxed{18\text{ minutes}}$.