My child was adopted by me when he was one and luckily these days we have access to lots of research which shows that the children who were told about their adoption before they were even able to understand it are the ones who have the least issues with it.
We are fortunate that because I knew this DS has always known and we have been able to talk through how he felt about it with me (I'm also a single parent).
At 10 we went back to the orphanage I adopted him from and he faced his past, met the current carers and childrne who were there and really had to face up to what his early life had been. Please don;t delude yourself that a 13 year old can't process the uncertainty about one birth parent when a 10 year old can face the grim reality of an orphanage.
I understand you're scared about him reaction and I do agree that you need to consider the right time and consider his "fathers" reaction however I have real life example of my aunt being told she was adopted in an argument around the same age. That's all it takes, one lapse in concentration in a fit of temper with your unreasonable teenager - and now there are at least 3 people who might let it out accidentally.
One of the storngest reasons apart from the research that it's better for a child to know early is that I felt really strongly that DS needed to be able to trust me completely. That I would nver lie to him and he could always rely on me to be honest and to support him in dealing with that honesty.
This is HIS life, you, your partner and your friend are all colluding in keeping a major event in your childs life from him. It really isn;t your right to do that and to find that the three people potentially closest to him have ALL lied to him will no doubt impact your relationship for a time. But leaving it too much longer will just make it worse.
DS and I both did DNA tests at Christmas - what are you going to do when he decides he wants one to see who his ancestors are... it's not an issue you can hide from these days.
There will always be a reason not to tell him - pregnancy, new baby, his birthday christmas, your birthday, it's a Tuesday. But there is one over-riding reaon to tell him - it HIS information, his life.