Hi, I am not an expert but posting in the hopes that this will bump your post up a bit - this board isn't the most frequented so you might be better off trying relationships (if you wanted advice more on the emotional side of things) or legal matters on the court side.
First things first, 'full custody' is not a thing in the UK any more, courts now determine 'residency' (who the child lives with), 'contact' (who the child spends time with and how much/when) and if the parents can't agree between themselves, other matters such as schooling, medical treatment etc. Crucially though, while the court can order the resident parent to allow the non-resident parent to have contact with the child, they don't have powers to make the non resident parent see the child regularly/consistently or otherwise be a good reliable parent, unfortunately 
. So I am not sure what going to court would achieve, your son already lives with you so that would be highly unlikely to change, the courts only order no contact to the non resident parent or removal of parental rights in extreme circumstances like abuse, severe addiction, criminal lifestyle etc. The prevailing theory is that it's better for children to understand where they came from, know and learn for themselves who their father (its usually the father) is and have some kind of relationship with them, even if said father is a total deadbeat/waste of space, as otherwise this can lead to emotional difficulties later in life, particularly in the teenage years such as dislike or denial of their own identity, poor self-esteem or also idolizing/fantasising about the absent parent only to have their hopes/dreams crushed when they finally meet the real person (and bear in mind in these days of social media it's alarmingly easy for young teens to track down and contact birth family).
You may or may not agree with this theory by the way but it's the one the courts subscribe to. So if you are hoping (to put it harshly) to get a court to cut your son's father out of his life, that's really unlikely to happen and it may even backfire and lead to more contact being ordered than he already gets, if for whatever reason he chooses to present himself as a loving and responsible father who dearly wants more time with his son (manipulative men are well able to deceive a court). That's not to say he'd stick to it of course but it could be a real headache/disruption for you as well as being upsetting for your DS. So if he currently doesn't have much contact and that's how you'd like it to stay I'd be tempted to not rock the boat too much TBH! The best thing is if you are able to remain on reasonably good terms with his father so that your DS is able to occasionally see him without it being a hugely disruptive or conflict fuelled situation - I know that's easier said than done and you must rightly feel resentful that he just dips in and out of your lives and then flits off to do his own thing without considering your DS. Does he pay maintenance at the moment? Is he named on your DS's birth certificate? Would he be open to agreeing between yourselves (and then sticking to) a more consistent contact schedule where he sees him for one Saturday a month or similar so you know to plan around this?
It might be worth seeking some proper legal advice from a family law solicitor just to help you understand the process and costs of getting a proper court order should you need to in future. One thing you could ask a solicitor about is granting 'parental responsibility' for your DS to your partner. This isn't exactly the same as adoption because his biological dad will still have parental responsibilities too (and will stay on his birth certificate etc assuming he's on there), but it means your partner will become effectively a 3rd parent and e.g. if anything happened to you would mean DS could continue to live with him etc.
As to what to do re helping your son emotionally, I think the best advice is to be as open and honest as possible with him starting right now (in an age appropriate way), answer questions when he has them and never ever keep secrets from him. Children are surprisingly resilient and open, and he will not be traumatised or upset by this if you don't present it to him as something dirty or bad/upsetting. Surely he must have asked questions already about his father if he does occasionally see him? Who does he think he is, what does he call him if he calls your partner Dad? 3 is the age where they start to ask questions about where babies come from and also where there start to notice things about families e.g. why does 'Jimmy' at nursery only have a mum not a dad, why does 'Peppa Pig' have a sister but I don't etc, so that would be the perfect opportunity to talk to him about the different kinds of families there are and what makes someone family. You can talk about how men and women make babies (age appropriate of course!) and how some people have a daddy that lives with them and some people have daddies that live somewhere else and they don't see them every day. You can talk to him about how mummies and daddies are the people that love and take care of us and that is why X (your partner) is also his daddy even though he didn't help make him, and so he is a lucky boy because he has 2 daddies (or if he doesn't call his bio father daddy you can say he has 'John' who helped make him and loves him etc even though he doesn't seem him every day and also X who is his daddy and loves him and takes care of him, whatever fits your situation and his understanding). So long as you don't show you find this difficult or upsetting, or in any way badmouth his biological father (whatever your personal feelings) it doesn't really matter how exactly you explain it, I would expect, to be honest his reaction will be 'oh right then' and run off/carry on playing, this is probably far more of a big deal to you than it is to him! Just don't wait and wait until 'the right time', talk about it to him as soon as possible, so that in the future there won't ever be a time where he didn't know about his father. There won't ever be a perfect moment when he is "ready", that risks it becoming some kind of traumatic 'secret', in fact the earlier you start making it part of the conversation the better and less upsetting it will be. He knows he is loved and safe/secure with you and your partner so he should be fine so long as you continue to reassure and support him even with intermittent involvement from his biological father - not saying it will always be easy, of course not, but he has the most important thing which is a family that love him!
Good luck!