OK OP, I think you've not had the answers you wanted so you've abandoned ship so to speak.
The others are right in saying you shouldn't get involved, it's not going to be as clear cut as you imagined, you might find things out about this 'sibling' you don't want / they don't want you to know. In the essence of keeping a professional relationship with the person it's best you take a step back, it sounds like he has a handle on things anyway.
I'm guessing he may have fought for contact rights when they allowed funding for family court cases, but I believe that has ceased. So he hasn't in my mind, spent thousands out of his own pockets going through the courts.
He hasn't been paying CM but he has managed to put money into an account for his DC. The account could have £5 in or £500. But seeing as he's been a student I doubt it's a significant figure. In reality the Mother needs money when the child is growing, not as a lump sum for them to blow when older.
He's helped you through tough times, so you want to repay the favour. You'll learn quickly that you can't really help or save anybody, they need to do this themselves. As the saying goes you can lead a horse to water, but it doesn't always drink.
You feel close to him, but we ascertained he's 22/23, the child is 6, they had the child when both were 16. Christmas has passed, did he not mention with excitement presents he was buying for DC. Ok he might not be able to get them to DC but it's proof if they get reunited, DC has always been on his mind. It seems like you've only recently discovered he was a father, which is odd for saying how close you claim to be.
You need to firstly keep a professional relationship with him, even if you perceive your colleagues to be like family. You then have your out of work relationship, whether it be a friendship or partnership. You can't let the latter encroach on the former.
I'm guessing you're relatively young, you think that you can save the world. You'll realise that dealing with other people's issues on top of your own is draining and quite often thankless. It would be something to avoid in general as others have said.
He needs to do the donkey work himself, it's unlikely that any charity / professional body will spend time with you, as they know that there's usually more to a story than being told. You're getting the shiny gold plated version, when in fact there are many variables. The mother could have wanted him to be involved but stopped trying as it was a waste of her energy. The mother could have been a victim of abuse from the guy, there's many different types of abuse. The families might not get along with one another.
If you excuse the fact he was young, why hasn't his parents sought visitation rights? Why don't they know where their GC is?
I'm not saying the above to be horrid or condescending, but to save you from a situation you can't really help.
An example, a family was split up after an incident where although a person was a victim, many knew the victim provoked the reaction. Regardless the other person hasn't seen the children since, in that time the victim has poisoned the children against this person. So they'll be hard pressed to ever really build a relationship. So not all men are at fault, they pay CM etc. Little can be done to help the person. That's going from knowing the circumstances.
You could spend your energy elsewhere and reap something positive. You can support this friend by listening to them and supporting them. But support doesn't always have to be action. It can be motivating them when they feel there's little hope. So keep supporting your friend, but as many have advocated, trying to save the day isn't a good idea. You will reach a point where you realise this.
I don't know if anyone has highlighted this, you say you've just gone through a horrid time. Isn't it better to focus your attention on yourself and your own recovery? You don't need to save someone else to feel good about yourself. Please believe me. If you are willing to give, there's always those who are willing to take. You seldom get thanks, you seldom get any recognition and long term you will feel deflated.
It's up to you whether you herd what has been said by many. But it's from experience. If you have feelings for the guy that's great, but don't go into a relationship thinking you can change the world for that person. I would seriously think about using Claire's law just to check that nothing untoward has happened for your own safety.