The key with questions like this is to really put yourself in the shoes of a trainee. What would be your first instinct? If it was me:
1. Communicate with both my supervisor and the partner. Explain that both pieces of work are urgent and you are unsure if you'd be able to produce both to the standard you would like. Ask if there is any room for an extension on either piece of work.
2. I would also see if you have any team members you can rely on to delegate some of the work to/help you out - this could be other trainees
3. I would then re-arrange my priorities for the day and finish the most urgent task accordingly.
4. Focus on client deadline driven work - client focus/client comes first (This is assuming that one of the pieces of work is client driven)
I think you should also provide an example of a time where I had to juggle deadlines on different pieces of work, just to show you really have the time management and communication skills these kind of questions test!