You suggest that
By the time next year...
Is syntax you have seen before. Using "the" in this phrase is ambiguous because it is unclear to what time you are referring. It is fully correct to say
By this time next year...
because "this time" clearly refers to the current moment of discussion.
However, the given sentence clearly contains "April". It is a good practice to avoid changing content when fixing grammar, so we should maintain the "April" because it may have importance in context.
The given statement
By April next year...
is unclear and possibly redundant. Saying
By next April...
prevents the year redundancy. You may assume that your audience knows the calendar and will recognize that April won't occur again during this calendar year.
The statement becomes unclear if it is currently before April this year. Pretend that it is currently March. Saying
By next April...
will suggest the event in question is only 1 month away.
In this case you should maintain fully disambiguated content
By April 2015...
TLDR: The given statement is somewhat confusing and redundant. Use "By next April" to be concise and clear.
Edit: There seems to be confusion about post positive adjectives.
By next April...
By April next...
Adjectives that occur after the noun they modify are not common in English, even though they are, technically, grammatically correct. Generally you want to place adjectives first unless you have a good reason. It is possible that your question leaves the post positive adjective because it is the grammatical change that least affects the flow and content of the original sentence.