I think it's just an ego thing, tbh.
You run the risk of breaking up two families, without even knowing if you are the father.
I do understand that you must have been gutted to be the unwanted one when you split up, and you had at least a couple of months of assuming you were going to be a father, so to have all that taken away from you must have been hard to take.
I'm not buying the 'I didn't pursue the paternity test because she said she'd leave" line though. It's frankly bollocks. If a paternity test proved the child was yours, you could have pursued custody. No solicitor that you spoke to (if indeed you ever spoke to one) would have told you to accept her calling the shots. Especially when you were paying them to determine paternity and subsequently custody.
And, um, why, if you were only doing what you were told, did you not go back to the solicitor again and start your paternity/ custody claim again when they actually moved away?
It sounds as though you seeing a photo or two has reinforced the idea that something that belongs to you has been taken away, and the old ego is bruised again.
This is purely selfish.
Write your ex wife a letter and give her your contact details. That way if she or the child want to get in touch, they can.
Can you afford uni fees? I mean, having not paid an ounce of maintenance along the way, I assume you have been saving it up for the glorious moment you are reunited, knowing how old your son is, what life stage he is at, etc?
Sadly they didn't want you then, they are unlikely to thank you for barging in and ruining their lives now.
I think you missed your chance. Too little, too late, and all that.
But do write to her and offer another paternity test and contact details if she ever wants to resurrect the debate.
Or, y'know, just become yet another fb/ friends reunited marriage break-up statistic.