DP has always shared the nights although in the early days with both DDs I generally did anything up til about 1am when I go to bed and he would do anything later than that. This suited us as I am the night owl whereas he gets up early anyway (generally about 5am). He is NOT good at getting up in the night but ALWAYS does it as he knows I really need his support.
With DD1 he would take his share of the hours walking her, driving her round, cleaning up the sick and all the other middle of the night tasks that she needed.
With DD2, who I really struggled with even though she is even easier than DD1, he has probably done more of the feeds, nappy changes and cleaning up of sick than I have in her lifetime.
He has had to be really supportive through the dramas of both our preemies even though he has continued to work full time (as do I when not on mat leave), has a75-90 minute commute each way to work and he still does bath time almost every single night. Then he will settle one to bed whilst I do the other.
He also puts up with me when I find it all so difficult - I would really have found it much harder if he had not been there to help through the nightimes when they were much smaller. I do think you need to talk to him about the help and support you need, especially if you go on to have a second child. If he doesn't wake to crying (from child) or poking (from you), poke harder! (DP almost always needed to be woken when it was his 'shift' and a baby was crying but he never complained about me waking him instead of sorting it - he knew we were both parents!