"一经通知就立即" can be translated into English as "immediately upon notification" or "promptly once notified".
Both expressions convey the idea of taking immediate action as soon as one receives a notification. Here are some example sentences for context:
We will start the work immediately upon notification. （一经通知，我们立即开始工作。）
Please respond promptly once notified. （一经通知，请立即回复。）