First of all, it is to note that English is a language that has its own dialects [variants] across the world; just like other languages, such as Mandarin, Arabic, French, Spanish, etc. The dialects of English can be British or American; Australian or South African; Indian or Pakistani . . . and the list goes on here on Wikipedia.
So here, you might want to ask:
Are American, British, Australian and Indian English languages or dialects?
I would regard them as both, languages as well as dialects. It all depends on the situation wether you refer them as languages or dialects. Note that the dialects of English language have hundreds of sub-dialects of their own. English is a language, of which American and British are its dialects. But in terms of putting the British English aside from the American, then both can be regarded as languages (in such conditions). That's all because the duo languages have their own distinctive sub-dialects being spoken or written in a distinctive geographical region(s). For instance, British English is a kind of language spoken widely in the UK (also in the Republic of
Ireland as per the comment of GEdgar left beneath this answer), whereas the Scottish, the Welsh and the Northern Irish are the sub-dialects of it, in terms of linguistic, ethnic, regional, and social lines.
Similarly, American English is a kind of language that has its own sub-dialects. See how people living in southern parts of the USA speak American English differently as compared to the North-USA-English-speaking people.
Besides, the Wikipedia site has something to say about the layout keyboards:
"The United Kingdom and Ireland use British layout keyboards, while Australia, South Africa, Canada, New Zealand and the U.S.A use American layout keyboards. In continental Europe English as a second language is nowadays sometimes even taught in American English, except perhaps in Scandinavia and the Netherlands." Link