14. I put my children's future as my highest priority. Their safety, security, happiness. Their education. 16. I work to live, not live to work
If she's actually cheating you'll have every right to divorce her and not bother reconciling or even trying to reconcile. Success can be found by leaving an abusive relationship too. That being said, divorce will negatively impact your kids (#14) so you may be amendable to considering reconciliation.
IF you are - it's a tough road to pursue and I suggest strongly that your likely best course of action should your wife commit to ending her affair and recovering your marriage with you would be to seriously consider the impact #16 is having on your marriage. Even the best marriages struggle when one of the spouses has to travel for work. Recovering your marriage and being able to hold your wife accountable as she defogs for months/years of wayward behavior and thinking would, IMO, very much necessitate you either quoting that traveling job OR moving back to London as I proposed earlier in your thread (which has the added bonus of moving away from the suspected OM).
I bring it up now as it may be something you end up discussing with her pretty soon here as a potential thing. Shortly after I busted my wife we spent many days and nights debating and discussing things. I did a lot of listening. Above you said you will "say what you want" but I found it important to try to say what's important but listen more. My wife wasn't ready for a lesson and I really wasn't going to teach her anything. She had to learn herself so when I did speak it was mostly trying to get her to consider another alternative, ask her to explain her flawed logic and listen as she went in circles and hit dead ends. Your wife isn't in love with you so your opinions, anger, threats and upset really aren't going to scare her - it'll just convince her that her current chosen path is the right one. Certainly the path from withdrawal to intimacy is through conflict but it might be a calm thoughtful and thought provoking conflict that is most productive towards accomplishing your goals (particularly #14). Be the leader of your family and, perhaps, by maintaining YOUR dignity and integrity you can lead her better to restoring and repairing hers IF & WHEN SHE CHOOSES to do so.
Here's a little quote I discovered.
One of the deepest feminine pleasures is when a man stands full, present, and unreactive in the midst of his woman's emotional storms. When he stays present with her, and loves her through the layers of wildness and closure, then she feels his trustability, and she can relax. -- David Deida, The Way of the Superior Man.
The internet is 90% complaining and entitlement and I hate it because I deserve better!