"We are British."
I believe you can say 不列颠诸岛 (bu2 lie4 dian1 zhu1 dao3) for "British Isles" (and 不列颠 for "Britain.")
My original reply is above; I have added the following in light of other replies.
While there is a distinction in English between "England" and "Britain," in Chinese, 英国 really does mean either "England" or “Britain," but not in every word or phrase in which English would use either term.
All by itself, 英 can refer to British (-something), English (-something), or can be an abbreviation for England or Britain (an abbreviation for 英国 ); 英 also means "excellent," "outstanding," "hero," "flower," or "blossom" (that "double cross," at the top, that looks a little like ++ , that's the "grass" radical that explains the potential "flower" meaning).
In conjunction with other characters, 英 can mean "British" or refer to "Britain." For example, 英军 , the British Army; 英属维尔京群岛 , the British Virgin Islands, 英国广播公司 (or 英国广播电台 ), the British Broadcasting Corporation (BBC), 英国石油 , British Petroleum (BP). Note that 英国 designates "British" in the Chinese names for the BBC and BP.
If you want to specify Britain (rather than England), you can say 不列颠 , as in 不列颠诸岛 , the British Isles.
If you want to specify England (rather than Britain), you can say 英格兰 , as in 英格兰银行 , the Bank of England, or even 新英格兰 , New England, which is actually part of the United States.
英国 can be an abbreviation for England (a shorter version of 英格兰 ), or can refer to the United Kingdom, whose official, full name in English is "The United Kingdom of Great Britain and Northern Ireland" (as of now), and whose full Chinese version is 联合王国 (but again, 英国 for short).
Accordingly, you really can say 英国人 to mean an English person or to mean a person of or from the UK (or, loosely speaking, a "British" person, but note that as a geographical term, "British" refers to the British Isles, not all of which belongs to the UK).
英联邦 is the Chinese word for the British Commonwealth of Nations (aka "Commonwealth of Nations," "British Commonwealth," or simply "the Commonwealth"). The British Commonwealth of Nations is actually much larger than just the UK; there are over 50 member states in the Commonwealth, nearly all of which were former parts of the British Empire, including Canada, Australia, India, and, again, about 50 others.
Speaking of the British Empire, 大英帝国 is the Chinese word for "British Empire."