Well DS' doctor thinks it is part of his ADHD, so maybe it's worth exploring these things for your DS. Not that that tells us what to do about it. But it is the basis for the referral to OT - he has since had an OT assessment, and they found that his core strength and balance is poor, which is probably causing some of the dysregulation because basically he's having to constantly expend so much more effort just to sit and stand, and it may be causing some of the frustrating behaviour like never being still, plus he wants to drape himself over people all the time or shove his face into your face or just BE very close to everyone, which is probably that he's looking for some deep pressure input. Knowing this is helpful, because we can try to give him deep pressure input (or, in less jargony terms, he needs daily exercise to get excess energy out and he likes to be "squished" to make him feel more secure). Although it will be more useful I think once we actually get regular OT sessions which we haven't yet.
We would not have picked up that he has poor strength - he can ride a bike with no stabilisers and a scooter and he goes climbing with DH and does well at this, and doesn't seem particularly uncoordinated or accident prone. So it was a surprise to be told that he struggles there. However, he does have this curious tendency which is when he's being really hyper and manic (and often annoying or silly as part of this) to accidentally hurt himself, which we always jokingly referred to as his "instant karma setting", because it's almost like the universe delivering some kind of instant payback - but I have reflected recently that I think what's happening is the more dysregulated he is, the less aware he is of where his body is in space so he is more likely to hurt himself as a result of this.
I also notice that he has poor interoception, which is awareness of internal sensations, so he doesn't know he needs to go to the toilet until he is bursting, he will not notice that he is hungry, or feeling poorly, he will not think to adjust his own clothing in relation to being hot or cold (he has only recently started to voice that he is hot or cold), and he doesn't notice emotions ramping up, they just all explode seemingly out of nowhere. This is also something we are hoping to work on with OT. I have no idea how to do that so we are more scaffolding it and trying to check in with him or let him know when it's been a while since he's been to the toilet etc. But this seems to be a big source of stress as often when he is in one of the "far away" moods, if we can get him to eat or go to the toilet, it helps.
I don't know quite how the UK system would handle it - perhaps posters in the UK could tell you. I know that 6 is about the earliest they assess for ADHD where we are. He was about 5.5 when I first phoned for the appointment and it came through about 8-9 months later. In the meantime I was lurking in a lot of ND parenting spaces and I picked up resources from there, like the idea of finding games which allow us to "squish" him, which I think is perfectly valid to do, even if your child ends up later not needing any assessment. If the "normal" approaches aren't working, it might be these approaches make more sense, and it might be a temporary bridge to help him in the short term or it might be that it's the start of a longer road. But it's definitely nice to feel less alone and less like you're the only one struggling with these things.