TKnow your rights. I suspected, and it sounds like, getting a D in the UK is going to work against you in terms of assets, custody, child support...etc. IF you can do something strategic or tactical to strengthen your case, you should absolutely consider that into your decision on how to proceed. Are you able to arrange things in your Luxembourg savings account to your benefit? Transfer it into the Arab Gulf state? ...etc, etc. Think about these things and consider your game plan. I know you don't want to get divorced, and you want to work this out, but if it comes to that (and that's up to her, not you at this point) you should put yourself in the best situation to get the best deal. Cynical? Maybe. But also smart. Better to be divorced against your will with my money and more time with the kids, than get divorce against your will with half the money and little to no custody.
To the OP, most of us live in the US or UK where the norm is 40-60% of assets and 50/50 custody. She's stated she doesn't believe she could even get a divorce without your permission. You repeated that, said she told you about the affair hoping you'd grant a divorce, but say there are other ways you think she could get a divorce, but you don't sound certain and she may not know of these.
I don't think you want her to stay with you because she feels trapped. In DB terms, set her free. The next time she brings up D, I would let her know you don't want one, but if she did you would grant permission and are committed to a fair outcome where you both see the kids and are provided for. (I would NOT spend any time working out what that looks like.)
Know your rights so you aren't fleeced. See a lawyer. Ensure she has access to one, too so she's not fleeced. The goal isn't the best possible outcome for you. In most countries, I do not emphasize this, because the courts ensure balance. If you want marriage to be a choice she makes because she wants to be with you, she has to be free to leave. In DB terms, don't do any legwork for her to divorce you, but also don't stand in her way (e.g., by withholding permission).
