Whats the resolution?

[Open Sauce]

In these situations, the woman holds the dominating hand. As she is the mother, she can make life as complex as she wants for her ex. It's a difficult situation but one that you knew you were taking on. I think that your partner should agree with his son, to see him for one day alone. In that time he could strengthen his bond with his son, and at an appropriate time he could point out that he's not happy with his son sharing his time with his mums boyfriend, but that is something he has to live with too.
Your partner needs to re-enforce his bond with his children, and needs to let them learn for themselves how important you are in his life, but it will take time.