DS3 had a tongue tie. This was finally snipped at 10wo.
He also had an upper lip tie. Like your DC, he had a gap between his top teeth, around the same, 3mm.
When he was 17mo, he fell off a chair...and split his upper lip tie. It looked horrific, like a bloodbath.
However - when his molars grew, they pushed his front teeth together. They are now perfect!
I also had the same, my gap was far bigger though - I could stick a pound coin between mine!! I never fell on mine (something like 50% of DC with an upper lip tie split it in a fall before their adult teeth come in).
Just after both my upper teeth came out, when I was around 7yo, I fell off a swing whilst swinging standing up, which I had been told a million and one times not to do... and split my upper lip tie. My top teeth are now so close together that I have trouble getting floss between them!
I do know that if you have an upper lip tie then very careful tooth brushing is advised, in fact I was advised to brush DS3's at least 3 times a day rather than twice. Also, milk in bed, in a beaker or bottle, or BM is going to be detrimental to the top teeth as the milk kind of 'sits' there. I was told to brush teeth after any contact with milk if they were going up lie down afterwards!
You may find that once he starts to find his feet, he ends up splitting it.
If you really want it cut, you will probably have to go private. As I found out when looking just before DS3 split his.