The childrens' natural father is the person who is responsible for paying maintenance for them. When you got divorced, there should have been a consent order stating what he should be paying in maintenance for the children, and for how long. You will have been given a copy of that at the termination of your case and if I were you, I would find it and see what it says, and take steps to pursue HIM for the maintenance for the children. If there was a maintenance order in YOUR favour from your first divorce, that would be terminated upon your second marriage.
At the time you divorce your current husband, you can make an application for maintenance for yourself under certain circumstances, but not for his stepchildren UNLESS he officially adopted them. It is worth bearing in mind that Courts are less likely to award maintenance to "homemakers" these days, taking the view that if one is able to work to support oneself and one's children one should be doing it. You may however, have a claim against his pension in the future, depending upon the length of the marriage.
During the divorce proceedings the Court will look at both parties' financial situation and contribution to the marriage, the length of the marriage, your ages, and each parties' ability to provide for themselves. He is under no legal responsibility to provide for another man's children, but he may be ordered to pay maintenance to YOU in certain circumstances. However, depending upon where he is working, if he refuses to pay it, it could cost you a lot of money to enforce any such order.
My personal advice to you would be to focus on the well-being of your children first and then yourself.
But be warned: just because someone "can well afford it" does not mean you have any right to make them pay for something they are not responsible for.