My mum only had one partner after my dad and I was introduced to him as a 'friend' immediately after my parents split so I think we know the timeline there. Did they really think I was that stupid?
She never gave a shit whether I liked him or his children and whether I was comfortable at his house. He was a permanent fixture immediately and was an abusive arsehole. I have never forgiven my mum for exposing me to this but in my case it wouldn't really have mattered if I'd met him after 6 days or 6 months. He was still an arsehole and she would still have stayed.
In my work I meet lots of families in difficult and abusive situations and whilst this may not be a popular opinion, people with kids have to be extra careful when bringing new people into their lives.
Get to know a partner slowly. Don't rush into stuff and really test them, the connection, how you work together as a couple.
Do you agree on basic stuff like attitudes to parenting and this is especially important if they also have kids.
How do they react when you tell them you can't see them, are they demanding of your time or do they understand that your kids will always come first?
Do a Claire's Law application. Find out what you can about them.
People get references, DBS checks etc for a nanny or a babysitter but we invite people we barely know to share our lives.
Talk about previous relationships, why did they end and how do they talk about their exes? Are they respectful?
When introductions happen, take it slowly and be led by your children's timetable and not yours/theirs. I was never once asked if I wanted this man and his children around all the time when I was with my mum (shared care).