My DP does loads and has done from the start. I think it's important to remember that dads don't 'help' or 'babysit'. They parent, just as the same as you do.
My DD is 8mo, but DP took a month off work when she arrived and did literally all the housework and cooking. He was also my chauffeur as I had a C-section.
Since being back at work, he takes DD every morning before work at 6 or 7. He works flexi-time so if it's been a bad night or he doesn't have anything important in his schedule, he will delay starting until 10 so I can sleep.
He doesn't generally do night wakings as she's breastfed, but he has done them when I've been unwell or when she's woken up soon after a feed so doesn't need fed. We kind of unofficially came to the agreement that I do nights a)unless it's a horrendous night where she's up every hour or something and I need his help and b) so he can take DD every morning and I can go back to bed, and that works well for us for now.
He also takes her when he comes home from work (literally, he walks through the door and is passed a baby
). He also does his fair share of housework. At weekends, he'll take DD for extended periods if I need to do something or just have a sleep. Yesterday, for example, he took her away at 7 and I got up at 10.
It's more difficult to do in those early months when they feed more often, but I used to express before bed so there was milk in the fridge in the morning for him to give her if he needed. Now she goes much longer stretches without needing milk. If baby is FF then he has no excuse.