He's really done a number on you hasn't he?
His happiness is not your problem, tough shit that he chose to have an affair, he got a second chance and didn't take it. Boo-Hoo!
As pp said if he starts with that , just says 'not my problem', if he carries on, 'stopped being my problem when you choose to cheat on me and continued to lie to me after I found out'
As far as contact goes, all you're responsible for is to have the children ready to go at the agreed contact times.
It is not your problem if he can't sort himself out a place to live etc.
It sounds very much as though you always had to be the grown up in this relationship, that girlfriend is not doing the same so he's expecting you to jump back in. (assuming that both she and her parents are aware of the children. . . .)
If he is a 'good' father he needs to start showing it.
Employ the nanny, email a suggested schedule of contact but reinforce that this will not be happening in your house (remember it's yours now, not his, he has no rights to it). I bet you £50 he complain that you're being controlling and you can't tell him what to do. . . just remember he can't tell you what to do either!
Yes the children might see less of him, but better that than growing up thinking that this is normal that it is ok for an adult to walk away from his/her responsibilities, and expect someone else to cover for them.
Very bad role model to show both female and male children.