The affair and you seeing the children are two separate issues, but the problem is devastated people are vulnerable and may well lash out, and that's imperfect, but very human.
Have you considered writing a letter to your wife stating that you understand her concerns and agree that it would be most destabilising to the children for them to meet someone else for at least a full calendar year, and that they need to adjust to the situation as it is without further disruption - and adding that you fully understand that this is an admission that doing so would be to place your own needs ahead of the children in an unacceptable way?
You could then put a second paragraph in stating that you feel that continuity and stability are essential to the children's well-being, and that seeing you in a consistent and dependable way will be key to their coping. You therefore suggest that you have them for tea every [Wed or other suitable evening] and every other weekend, weekend of choice to be determined by her. As the 2 year old is still so young you appreciate that it should be just Saturday night at this stage, unless she would prefer the break, in which case you are happy to have them on Friday and Saturday night.
Then reiterate that you want things as calm for them as possible, and you are fully guaranteeing that there will be no contact of any kind with any partner for a full year. You simply want to secure their relationship with you, because you love them, and because you know that it is essential for their future emotional health that you do.
Best case scenario is that she is reassured by this and starts to relax. Worst case scenario is she doesn't, but you have clearly demonstrated good faith and an ability to put the children first should things ever reach a court stage - which obviously is a horrible thing to happen for all concerned.
I'm not touching the affair part. I will just say that walking out on someone who does not meet your needs when there are 5 year old twins and a 2 year old is shitty conduct, and she probably hasn't the spare energy or time to meet anyone's but the kids' right now. I wouldn't either.
But that is separate from the need and right of the children to have a great relationship with their dad, and they need to know that you love them and that you are a part of their lives, and not going anywhere. And she needs, sadly, to find a way to deal with that, whatever her emotions right now. It's going to be agonisingly hard for her, and she will need your patience and reassurance that you aren't going to compound things for her and the kids by bringing the OW into it, but she does need to find the strength from somewhere to do that.
Bring up mediation with her again in the letter, saying you see it as a way to ensure she has her worries heard and addressed in the best interests of the kids, with a neutral third party there to protect her and their interests, and that you hope she's willing to attend on that basis.
Her inevitable answer will be that you weren't thinking of the kids when you decided to have an affair, and she has a strong point. You weren't. But it is what it is and all you can do now is try to exercise damage control.
I wouldn't touch her behaviour in front of the kids, unacceptable as it is. She's not doing it for fun, and the hope is that in time she will stop as she feels less raw. If things have to reach court, then that is when you need to voice it. Meanwhile I would keep a diary and email it to yourself daily with the diary as an attachment, so you can prove you are writing it contemporaneously and not, later, to give yourself evidence that she's unreasonable.
