I'm so sorry about the "London" thing; I am sure there will continue to be other "shoes" that will drop. Once you get over the initial shock of the actual affair, it really ends up being where the DECEIT and the DISRESPECT bother you the most, as strange as that may sound.
I hope you are documenting everything. Wouldn't charges from that show up on your debit card or your Amex account? I'm wondering if she has another cc, in her own name, that she has paperless billing on? No matter; in any even she will have to do a FULL financial disclosure, under risk of perjury, as part of any legal action should things come to that. Judges don't look too kindly on squandering family assets on affairs, ESPECIALLY when there are kids involved.
As for doing stuff as a family, I would keep it about 90% with you and the kids on the weekends. Maybe occasionally "Hey, I'm taking the kids to _______, if you'd like to join us we're leaving about 3." Very casual and almost business-like. If she joins you, great; if she turns you down say "No problem. We won't be gone long" (or whatever). Do NOT beg her or pursue in any way.
Also, be wary that she doesn't use those family times to be texting or phoning OM -- that will be a boundary you will have to lay early and firmly.
But make sure you absolutely schedule some things with JUST you and the kids, as this is one of those "reality" jolts she's going to need to learn to live with. If she's going to basically fire you as her husband, these will be some of the realities she will have to grow accustom to, and the hope is that when these things happen now, she won't like how they feel and will make a move back toward her family.
