It is unlikely that a water leak would require a call the emergency services, such as fire or ambulance. However, were the leak to be a risk of causing electrocution and you could not locate a safe way to shut it off, you could find you have little option. This obviously would only be in dire need with an immediate threat to life and not because you could not find the stop cock!
What is important is to remember that there is a duty to “mitigate loss” from an insurance point of view. You are expected to take action and possibly turn off the whole water supply to stop water damage spreading. For a limited time, the inconvenience of no water is not a reason to leave a leak running.
From a water leak detection point of view, we find most visits are ‘urgent’ but not an emergency, and as such, we are not “call out plumbers”. Our role is to identify the leak and provide support through the process, normally as part of an insurance claim and including mitigation drying and the claims process as a whole.
Total flooding is obviously an emergency, and services such as police and fire brigade along with your local water company will likely already be aware. In these cases, the health risk is substantial as flooding will normally cause the sewer to back up and there with be waste contamination to contend with. This will often cause the failure of the electrical system in your home as the water effects the electrical outlets in the home.