This is a tough spot to be in. I've been in this situation and have tried both scenarios.
Working with the Other Manager:
In the situation where you work with the other manager, you would let him know what tasks you need completed and ask if his team can commit to those deadlines. In this scenario, your skills of persuasion are important. The other manager must have a stake in the project. If he has a reason to succeed as well, then you'll be more likely to be successful yourself.
However, if he doesn't share the same sentiments as you do regarding the importance of the project, and if his goals are focused elsewhere, then you may end up approaching the deadline without the completed components you need.
If you think that working with the other manager may not prove successful, I suggest you sit down with the stakeholders and determine what the most critical functionality is, and then have your team work on that part. At least then you have control over your destiny!
Take on Responsibilities
If you know that you absolutely need to have the work in question completed, you could volunteer your team to handle the responsibility. The main drawback here is that it may create other delays on your team because they'll be focusing on double the work.
I would suggest selecting this path only if you're sure the other manager won't be able to meet the deadlines. This will create a burden on your team, and you may need to put some functionality on the back burner if you go this route. Your chances of meeting all of the requirements does drop in this scenario.
In the end, you may find that you and your team are overwhelmed and have missed the deadline.