I don't think it sounds like ADHD or autism, well not to me anyway but I am not an expert on it.
If he has a weak core and low muscle tone I would think it was more likely his fidgeting and squirming around is more likely to be something like proprioception related issues/sensory stuff?
I have a 7 year old and a 5 yr old. neither sit still when at home but do at school apparently. the elder one in particular always does a half sitting half standing thing, that is the closest we can get to sitting generally. She has excellent core strength and muscle tone but a lot of dyspraxic type tendancies. She has poor proprioception (is hypermobile but due to her good muscle tone etc they won't actually diagnose it for some reason) apparently and they told us she probably feels the need to have her feet on the floor. Is his chair and table a good height for him? does he need a foot rest? does he need something to lean his elbows on? is he better if he sits on a cushion or a wobble cushion or something I think they are called. I think it means they can wiggle around on it and get more comfortable and less distracted.
social interaction can indicate a problem but it can equally just be maturity related or even just confidence.
It also isn't unusual for a child that age to get so engrossed in something that they ignore everything else, lots of children forget to go to the loo because they are busy playing/drawing etc and end up wetting themselves. It is again a maturity thing which is why it is hard to identify problems in young children.
Did the physio give any other help? exercises, ideas, info, other comments which might indicate a problem? how is he coordination wise? dyspraxia is a very wide ranging problem. very hard to get diagnosed in our experience unless you tick certain boxes out of their list but the tips and techniques for dealing with it can also apply to lots of people who don't have dyspraxia but have bits of the problems.
I would start trying to build his muscle strength and core stability to be honest, don't worry too much about the school work. it is the holidays shortly (well assuming you get a break over Christmas, depends what country you are in I suppose). We use Wii balance games with a balance board, we also have a musical wobble/balance board, hula hoops, simple yoga, ballet/swimming/football are good too. the physio should be able to give ideas of exercises.