If they ask you to work again I would say unfortunately I gave to decline. I've been told that you are telling my clients I am always late. This is my career and my name and reputation mean so much to me. I don't feel comfortable working for people who would lie about me and try to harm my reputation.
I wouldn't do this. Why engage with gossip? You don't actually know if they said that or what they said or to whom. You just know what someone else told you and you don't even know if the person telling you is trustworthy. Honestly, you are making something out of nothing. If you don't want the income from the other family, just decline the job and move on. Your reputation is built on what you do. Keep showing up on time for the jobs you take and you'll be fine.
Oh, by all means, let the lying MB off the hook. She wi continue to lie..
You're missing the point. OP doesn't know if the MB is lying or not. She only knows gossip from another parent. She'll look pretty stupid if she engages in the gossip and turns out to be wrong. She could also damage her own reputation by confronting the MB if she is wrong by being known as a crazy sitter who makes outlandish accusations. Besides, does anyone believe that the MB, IF she did say that she was late because her nanny was late, is actually trying to harm OP's reputation? That's sort of foolish, especially if she is still hiring OP for sitting gigs.
If OP is uncomfortable with this MB because she might have said she was late, all OP has to do is turn down any babysitting jobs with them. If she rises above petty gossip and does a good job, her reputation will speak for itself.