Today is Tuesday of week 1.
I have just been told to do X until next week.
When is the last day I can do X?
Sunday of week 1? Sunday of week 2?
According to Wiktionary, Until basically means:
Hence, accordingly, until next week means, before next week begins, i.e., the Sunday of Week1.
There are several problems with the question.
This means carry on doing some continuous work named X from now to [some unspecified day and time in] next week. Then stop doing X [forever or until instructed otherwise]
Means let me know the name every time necessary from now to [some unspecified day and time in] next week. After that, don't let me know the name.
This is an unusual construction and might be used if the requester is forgetful or if the name changes many times each work - for example if it is the name of the winner of the daily crossword and the requester has to do something with that name every day (print in tomorrows newsletter, send prize ...) but will not be doing it next week (vacation?).
Means I don't need the name now, but I will need it next week [probably any day and time in that week]
Means I can wait but I need it sometime between now and the start of next week.
There is inherent ambiguity about the start of a week and the point in the week that would be acceptable when you ask that something be done "next week", you should expect rising anxiety in both parties as the week progresses.
I would interpret it like this:
Sometime unspecified time during week 2, you'll be taken off of X and put on a different task.
However, many non-native English speakers confuse the words until and by and say something like this when they really mean they want you to finish a project by a certain time.