"Nigerian English" is a term colloquially applied to two classes of varieties, but it linguistically applies to only one. On the one hand, Nigerian Pidgin English, a pidgin with all the characteristics of a pidgin, may be referred to in the shortened form of Nigerian English. On the other hand, Nigerian English, which is the L2, or second official language, of Nigeria, is a variety of English developed from elite educated Nigerian's learning and using English.
The background of Nigerian Pidgin English (NPE) is that it developed originally in the 18th century as a lingua franca between traders and later between missionaries and converts. The background of Nigerian English (NE) is that during British colonization, English was learned by elite educated Nigerians and used in administrative employment. Presently, according to Funso Akere, NE has reached the status of a native language, or mother tongue, because generations have been born for whom English is a language spoken from infancy at home (the same is true for varieties like Indian English and Malaysian English). Moreover, NE is used for formal education (also as is true for IE and ME).
As with all varieties of international English, including American English, there are phonetic (sound) particulars that have been assimilated (influenced) from indigenous native languages and vary from the original English English, though the phonemic features (units of meaning) are no different. Also typical of all varieties, lexical (vocabulary) differences result from borrowing lexical units from indigenous or other prevalent languages.
Whichever meaning of "Nigerian English" one uses, whether the shortened form of Nigerian Pidgin English or the L2 Nigerian English, both certainly exist. There are regions where only NPE is used alongside indigenous languages such as Igbo and Yoruba; it certainly exists. In formal education NE is used; it certainly exists. There are two most likely possibilities for the future of NPE. One is that it will expand into a creole and then become another variety of Nigerian English. Another is that as education expands to remote areas of Nigeria, NPE will be replaced by NE. The least likely possibility is that NPE will remain static and keep its current place in the language mix in Nigeria. Since language is by its nature dynamic instead of static, this last possibility is less likely over an extended time period.
Personally, I think "Nigerian English" is a valuable version of the language that so many of us share. I have many friends from Nigeria; I appreciate how they speak (that is, their accent) and how they share their local cultures/languages when choosing to speak in English. There are many excellent Nigerian writers, like Wole Soyinka; but then, there are many excellent Nigerian speakers--at least among my friends.
Nigerian english or "broken" certainly does exist. It is a wonderful gift to be able to speak what I consider a language akin to the so-called "ebonics" or the at home way of speaking for many African-Americans in this country.