How to deal with a busy research partner?
As the comments have said, you need to go with your partner and explain this to your adviser face to face. It's not an incompetence issue and you need to make sure that you make that clear to your partner - it's a workload balance issue. 

At the end of the day though, there are only two options available:


Either your partner reduces their outside workload or,
Your adviser reduces your workload to account for it.


This is something that needs to be decided between the three of you. 

However, I'd strongly advise that after you've had the talk, that you put in place a very rigorous project plan for your next tasks with a strict timeline. Often, this situation ends up just repeating itself as your colleague says 'oh I'll find more time' or your advisor says they'll slacken off but not much happens, so a rigorous project plan helps highlight if the talk has worked or not.