I'm perplexed about where your Dad is in this plan? You mention him possibly prompting a daily walk, but that's it. Don't they live together, and presumably share many of the same healthy or unhealthy habits? Don't they eat the same meals?
I see your Dad as being key to the success of your plan. IMO, you need to start with getting Dad on board to make some healthy changes as a couple. I don't think it is likely to work if you single out and embarrass your Mom about her "problem." And it undoubtedly will embarrass her, no matter how gentle you try to be.