Emergency numbers differ by country, so the honest answer is: look up your specific destination and save the numbers in your phone on day one. That said, a few patterns cover most of the planet. The number 112 works throughout the European Union and in a surprising number of other countries, and on most mobile phones it redirects to local emergency services almost anywhere. North America uses 911. Many countries in Asia and elsewhere run separate numbers for police, fire and ambulance rather than one combined line; India, for instance, has consolidated services under 112 alongside older dedicated numbers.
Tourist police are a separate and underused resource. Dozens of tourist-heavy countries operate dedicated tourist police units or hotlines specifically staffed to handle foreigners, often with better language skills and more patience for visitor problems like scams and theft.
As for round-the-clock English support, do not count on it universally; big-city dispatchers often manage, rural ones often do not. Your fallback chain is worth memorizing: hotel reception can call and translate, your travel insurer's assistance line operates in English around the clock, and your embassy runs an emergency line for citizens. The country's official tourism website lists the exact numbers.