I agree completely with what the others have already said.
I and BPDw are separated and share custody of S8 50/50. He doesn't have much responsibility at BPDw's. Sometimes I think it's a free-for-all over at her place. I have some chores for him to do each day and some more chores for the weekend. If he gets right to them, the daily chores only take him about 5 minutes, and the weekend chores take him about 20 minutes--it doesn't seem like much, but it helps me a lot!
I have S8 keep his bedroom and bathroom picked up each day, and fold clothes and empty the dishwasher whenever I do laundry or run the dishwasher. He does these very well. He's learned that it doesn't take him long when he get's to it. I also have him vacuum his room and other rooms every now and then. He's very self-sufficient with all of these.
Last year, when S was 7, I had him pick up his bedroom and bathroom as part of his morning routine, and I had him fold clothes on Saturday. As long as BPDw didn't interfere or distract him (she was so good at that) he did very well with it.
My answers to your question:
From my experience, yes, the kids will be willing to help out, even if BPD partner doesn't.
As for how to encourage them, they won't need that much encouragement once they get used to it. They will get used to it, as long as you are consistent about requiring it. Your children will learn that this is what you expect from them.
More tips: first explain to them why it needs to be done. You could also tell them that you need their help. Give them hugs as a reward (I need to do this more). I use a sticker chart. For every sticker that gets put on the chart, S8 gets a sticker he puts wherever he wants. He used to put them on him--then on me, but now he puts them on other things, like his door. You could add encouragement by doing something with them after they finished.
Also, don't expect perfection, at first, praise any effort, even if it is far from perfect. You will probably have to show them how and maybe even do most of the work (some of their encouragement will be getting to spend some good time with you). It may even take you more time at first. But, after a couple of times they will be able to do it by themselves (if the task is age-appropriate). If you want it done better next time, perhaps wait until next time and show them how to do it better at that time.
Mostly, though, not much encouragement will be needed, once they get used to it and understand why it needs to be done. They will be getting a lot of self-encouragement through the self-pride they will get by taking care of themselves and their things and through helping out as part of the family. They will see themselves as part of the "team."
If they ask about your BPD partner not doing things, I suggest not really defending BPD partner, but not putting them down either: Instead, just say something neutral like "I don't know."
In my view, it is important that your teach your children to do chores is important to their emotional growth. They need to learn that doing chores is a necessary part of life--and not something that they should be paid to do, or even something that they should be excessively (sp?) be praised for (eventually, they need to learn to praise themselves).