Give yourselves lots of time before introducing a new adult (and her or his children) to your children.
When we’re in the honeymoon stage of a new relationship we often think, “ because I like Dave I know my girls will.” Be sure you’re past the honeymoon stage of the relationship before introducing Dave as a potential new member of the family. Be cautious of quickly filling the hole left by the last relationship with a projection onto the new partner of someone you believe will dispel your fear and loneliness.
In summary:
Let your kids talk on and on and on about how much they wish you and Mummy/Daddy would get back together. Let them see you cry about the end of the first family. Let them cry and rage. Let the grief process happen.
Have a conversation with your kids about whether they’re ready for a new step-parent.
Give your children and your new partner full choice of degree of closeness. Just because you’re in love doesn’t mean they will be. Don’t push “happy families.”
Give your partner full choice to not take on any caretaking roles if they wouldn’t be fulfilling and meaningful.