I am so sorry Op that you and others like me have to go through this .
You will get lots of great advice from those who have experienced the level of shock and pain and other well meaning advice from those that haven't . Sadly you will not know how you will react until you experience it yourself .
Each relationship is unique with your own family set up and strengths and weaknesses .
You are currently in shock , your world as you know it has just imploded .
The most important thing that you can do first is to look after your own well being physically and mentally . You need to prioritise this as otherwise you will not be able to look after your children . Do not beat yourself up - you did not cause this . You were in the dark until recently. Your H was aware of what had happened and was able to deceive you for a sustained period of time . Be kind to yourself , concentrate on feeding and looking after you and the kids .
As others have said I think you should get individual counselling . You may be able to access some through your work if your Employers are signed up .
It would also be useful for your husband to get individual counselling as well and then if you are both still willing - to then undergo joint counselling .
Also lean on your support network , do you have family / friends who can help out or at least be available to speak to you on the phone ?
I think it would be helpful if your H was able to stay elsewhere to give you some breathing space but you will have your weigh up re childcare if this will be better for you or not .
I think in your situation a lot of people's instinct is to stay together . If you could turn back the clock to before this happened , who wouldn't want to do so ? You are in shock and pain and this is the "easiest " position i. e the status quo.
You do not need to make a decision quickly . Take your time , even if you come down one way , this can be changed . You need to do what is best for you and your children .
People can survive affairs if they are both willing to work at it . The emphasis has to be on the cheating partner doing most of the work . However realistically you will need to be prepared for the odds being against you .
I say this with kindness as I don't think it will be helpful for you long term not to be aware of the blunt facts here and why it will be difficult for you to stay together . Difficult but not impossible .
Your H was able to continue to cover up & continue the affair for 2 years . I think it was clear that he and the Ow got a thrill from the illicitness of this . This was not a drunken one off . It also looks like it was the OW who called it off and was uncomfortable with the situation - not your H .
Only you will also know your H's reaction and remorse . An important factor is why did he tell you when he did ? What was his for motivation for doing so . Was it guilt or his feelings for the OW . Sadly even if the OW does not want to be in a relationship with him , what would your H want . If she was open to this - what would his honest answer be ?
What is your H prepared to do to repair your relationship ?
I know it will be painful but I think you need to know what you are dealing with , get a timeline from him and also ask for access to their messages etc .
Finally , I know that you have also said that you don't think you should speak to the OW's H . I appreciate that you said they are now separated , but I think it would also be beneficial for him not to be kept in the dark . Additionally he may also be your ally in piecing together the facts and also may alert you to any possible reigniting if the relationship between your H & the Ow .
Statistically also when your H has already cheated , he is more likely to do so again .
I wish you and your family the very best for the future , whichever way that turns out . Look after yourself 
.