In America, the most common language for deaf people is American Sign Language. In fact, it's the fourth most commonly used language in the United States!
Despite the fact that English is spoken in both the United States and the United Kingdom (accents and regional dialects aside) there is an entirely different sign language in the UK from the sign language used in the United States. British Sign Language (BSL) is notably different than American Sign Language (ASL). There's also a French Sign Language (FSL). There is no universal sign language, and different countries or even different regions have their own version of sign language. ASL is also expressed differently in different parts of the country, giving the language as much of a variety as one would expect from any spoken language.
ASL is also completely separate from English. It has entirely different rules for grammar, punctuation, and sentence order.
Check out this link if you would like to learn more about American Sign Language.