I feel so sorry for you. I would be getting out the OW Voodoo doll if I were in that situation, and to Hell with anyone who thought I was evil.
You did not create this situation, and yet you are expected to be the mature adult one and deal with it.
I would let the kids meet OW once before the baby, but not on their own. I can't believe what bollix I have read about "their father being the equal parent" and "he is capable of being responsible for them"
What??!!
He is not capable of putting a condom on his dick whilst having an affair, so I really wouldn't trust him to make the best decisions here.
No way would I EVER let my children spend time with someone I had not even met.
You need to meet her. Keep your cool when you do, for the childrens sake. Be business -like If you can, but you need to check her out.
Then let them meet the half sibling later, and play it somewhat by ear. After all, by then they might have split up again, who knows.
I agree with mathanxiety(way back there..)
This OW will not really want your kids around anyway, she will be too busy with the new baby and trying to put her stamp on your total arsehole of an ex husband.
Try not to panic, try and do all of this without the intervention of Family Court, for the sake of your kids, but don't be rushed and don't be bullied.
You will feel better, and it will get sorted out, it will just take time, and at some point you will meet someone deserving of a loving wife and family.