The financial resources of the new partner are relevant, but only indirectly. It is not possible to demand that the new partner has to pay income or capital to the former spouse.
The court can only make orders against a husband or wife of the marriage in question. But if financial resources are being shared, then it does have a knock-on effect on how much can be afforded.
For example, if a husband is sharing the household bills with his new partner then clearly he can afford to pay more by way of maintenance to his wife or to their children.
Similarly, if a wife has started to live with her new partner who already has a home, then arguably she does not need as much as she would otherwise to buy a new home. Dividing the assets may be simpler or more difficult depending on the case. But the person cohabiting is likely to do less well.
There still prevails the myth of common law man and wife. Increasingly in the 21st century couples live together as if married but without going through a formal ceremony of marriage. Many believe that after a certain length of time, they acquire the same rights in law that apply to married couples, both while they are together and upon the dissolution of their relationship.
Unfortunately this is a myth. For example the rules of intestacy and relating to the next of kin should one party die, do not apply where a couple is not married.
You cannot simply stop making payments. If you feel that these new circumstances merit a reduction in payments then you need to apply to the court for a variation to the order for spousal maintenance. If the court agrees to vary or stop the maintenance it can backdate the change to the date that you applied for the variation.