I think I would allow her to set the pace on this. New mum's don't take kindly to anyone else meddling with their routines and way of doing things. I know I didn't. Most of the time what people thought was helpful made me want to tell them to fuck off and leave us alone
so, ya, a quick telling her that you are available, don't want to be a burden, but would be around more to help if she is willing would be enough. Then drop it.
Dad's are kind of not that useful when babies are born. Their roll is more of the go getter, the meal maker, the shoulder to cry on, the laundry doer... and the chauffeur, cause it's firstly all about baby and secondly all about Mumma. About the only thing that is for Dad's at that point is rights to walking around with strollers looking proud of their off spring. Hew will have his hands full.
As for you and this man? I think you are heading into some potentially rocky times. I think he is very wise to have the forethought that he is not going to be very available to you emotionally and time wise. I think that preparing for his absence from your life is probably the best you can do as his whole world is going to change and yours will along with it. It's hard to say where that will lead, but I would think it would be better to be pleasantly surprised when he is available than to be down and sad because he isn't.
It concerns me that you are thinking this relationship has become more than what it started out as. Have you and he talked about this? Does he agree or are YOU feeling this way? If you haven't talked to him about it, I suggest you don't and let this be his move. He is coming up on a HUGE responsibility and the last thing he needs is the woman he has had a friendship with and some cool sleepovers with changing the game on him. You agreed to the casual nature of your relationship and he could very well be done with that in a very short time after baby comes. Prepare for that realization in him. He could either be done, or be just coming over to get laid... both of which are not serving your best interest...
The thing with these casual things is that emotions and expectations and assumptions sometimes get in the way. I had or potentially still have a casual intimate friend that I used to see regularly, but he is going through a divorce and has a mono girlfriend right now and so I have been told he doesn't have time. I am hurt by that. I was there for him during a tumultuous time in his life (several times) and now that his divorce is going through he has no time for me and prefers to spend his time with his new girlfriend, who, btw, was okay with my being in his life, it's him that chose to end that part of our relationship to concentrate on his mono relationship.
The result has been that I made myself available during rough times and when they were over he was done with me for now... My question to myself is, will I be so available next time? Or should I be very careful not to be SO available. This is the second time around for us, where I have made myself a fool for this man and I feel used and unappreciated now as a result. Not his fault, I agreed to the casual part, but now I need to be sure not to d that again, because he means more to me than I do to him.
I would suggest that you make sure that you are getting your needs met. Not necessarily with him, but with other men that are more available. I don't know the story of the rest of your life, perhaps you have stuff going on elsewhere, but a word to the wise.... not that he is a bad man, but he has got the best thing ever in a woman to have sex with while his wife is huge. Most men have to suck it up for years before having sex with their baby mumma's again and this man has you... be careful that he isn't taking advantage of you and that you are getting some other needs met from him or elsewhere rather than turning into a nice fuck every now and then with nothing more... you are worth more and deserve more. If you are okay with helping them out by being the object of sex then great, but it sounds like your idea of moving forward is to mean more to each other than that... be careful with your heart and good luck.