Introducing girlfriend to kids - timeline?

6 months in with Sarah and it's getting serious. My kids (11 and 13) have NO idea she exists yet. They still think I'm just "going out with friends" on her weekends. I know I'm being a coward about this.

Ex found out last week (through the grapevine, how embarrassing) and sent me this message: "I'd appreciate a heads up before the kids find out from someone else." Which... fair point.

Timing feels impossible. Their mom is still pretty bitter and I genuinely worry she'll poison them against Sarah before they even meet her. But also, lying by omission is not great either.

How long did you guys wait? What's the right move here?