イギリス sounds like English, but actually イギリス is NOT equivalent to English. According to this web page, イギリス is an import word from Portuguese language. It originally means England, but its meaning has changed in Japan. It doesn't only mean England, but entire land of the UK now.
So, イギリス is the equivalent to the UK.
イギリス人{じん} is equivalent to British people, usually means people who have British nationality.
イングランド is the equivalent to England.
イングランド人{じん} is the equivalent to English people.
By the way, there is a Japanese word [英国人]{えいこくじん}. It sometimes means English people, sometimes means British people. So you should avoid using it if you want to make it clear difference between English and British.