You say your husband is great with kids, and he is, with kids he can really interact with. You've seen him with his 12 year old and other older kids. Many people have trouble interacting with very young children. After having my first child I found it hard to really play with her myself and I was her mother! They are almost subhuman for a while. I don't think he favors his first child over his second, it's just that he doesn't KNOW his second child because the baby doesn't have a personality yet. It's possible that with his first marriage things went the same way. His first wife probably took care of most of the baby's needs and he took up the reigns when the kid was old enough to respond to him in a way he could work with.
The relationship between my husband and I did change a lot after the birth of our child but it does snap back after a while. It took us almost three years though to grasp a bit of what we had before the baby was born. It will never go back to the way it was before completely, because things are different now, but they do get better.
My husband will walk over the same pair of dirty underpants on the floor until they solidify and have to be removed with a chisel if I don't say anything. I know it makes you seem like a nag but some people, not just guys, are not as observant and considerate as others and at his age, he's pretty much cooked now.
So, knowing this, you'll just have to work around his limitations and get better at asking for what you need and want and realize you're not nagging but helping him to help you because it's all together possible he doesn't know any other way.
If you want him to do something, set it up in advance and if he doesn't follow through tell him it hurts your feelings and makes you feel terrible that he seems to ignore you when you ask him for help. If he helps when you ask him to, make sure to thank him and tell him you appreciate his help. People don't normally like to do things unrewarded. There are very few Mother Teresa type people born to this earth who would do things for others because they just love helping, so make sure to reward behavior you want to see repeated and admonish the behavior you would like to curtail.
You be his eyes and ears since he doesn't seem to notice when things need doing. Then just politely ask him to do whatever thing you need with a smile in your voice and be honest with him. I'm positive he's not actively trying to make your life harder or doing anything on purpose. Some people just work differently than others and you have to work around the quirks of your mate.
My husband's tolerance for dirt and filth is a lot higher than mine. He can let things go in the house a LOT longer than I can so when something starts to irritate me, he's perfectly fine with it and willing to let it go longer so it's up to me to initiate the cleaning and rope him into helping since if it were up to him we'd be ankle deep in our own waste before he was phased.
Just don't think of it as nagging. Think of it as asking to have your needs met in a positive but assertive manner. If he doesn't meet your needs, tell him your needs are being back burned by him in your estimation and that you really could use his help with the upbringing of his new offspring. I really don't know his personality... or yours, so it's pretty hard to give advice tailored to you.
I just know that my guy is a sweet, caring, loving man who is thick as a brick when it comes to what I need to feel content most of the time but I've slowly trained him to meet my needs over the years by sweetly asking, praising heavily and puffing up his ego when he takes care of what I need taken care of and showing my disappointment and sometimes using tears, not anger, to show him that I'm unhappy if he ignores what I need him to help me with.
Also, I know it seems a little underhanded and devious, but rewarding with sex can go a long way to cement in the mind of your man what things you need him to take care of. It's a kind of unspoken, if you scratch my back, I'll scratch yours kind of arrangement. Right in the middle of foreplay just thank him again for whatever it was he helped you with that day, regardless of how trivial it was. It'll stick in his mind and he'll associate that chore with his good feelings.
It worked for me.