I think the “until when” construct can work fine in a question like this, although I think I’d be inclined to change the order of the words:
I have until when to finish this project?
Also, it’s worth noting that we will often put additional emphasis on the word when in such questions, particularly when expressing surprise. For example:
Ted: I’ll need this project done by Friday.
Ned: I have until when to finish this project?
In this dialogue, Ned isn’t really asking Ted for a deadline; he is expressing incredulity that the deadline is coming so soon. In other words, Ned knows the project needs to be done by Friday, but he is astonished that he has so little time to accomplish so much work.
Of course, there are many other ways people can ask about an upcoming deadline. I especially like the suggestions provided by Bob Jarvis and Ronnie Childs elsewhere in answers and comments under this question:
- When is the deadline for this project?
- How long do I have to finish this project?