The ex husband will have no financial responsibility to the woman whatsoever.
The natural father of the children is financially responsible for them.
Why isn't the woman financially responsible for herself? She is an adult after all.
However, she will not be able to claim means tested benefits in her own right as her new husband's income will be taken in to account.
If they subsequently divorce, the second husband MAY (not automatic) be required to pay his ex-wife maintenance.