Troubled Grandson

Btw, my DGD did also scream and/or cry on seeing me.

My DGD, now 4, used to try to avoid me when I visited. I live quite far away so only see them 4 or 5 times a year.
I didn't try to entice her or interact with her at all, but just talked with her parents and she'd gradually get used to me being there.
Now she is a little older, things have improved.

Along with your DGS's behaviour at nursery, it sounds as if he isn't yet sure how to interact with others.

Just let him go at his own pace.

Could you try ignoring GS? Maybe if you don't look at him or speak to him he'll eventually approach you. Could you sit on the floor and examine a toy or book you've brought as a present for GS? However unreasonable it sounds, he may feel threatened by your presence.
I was about 6 when I used to hide in my bedroom when visitors arrived. I mean perfectly pleasant aunts and uncles whom I didn't dislike but didn't see very often.

Thank you for your responses. I am a grandmother 6 times over and this is a first to me. I get on extremely well with sil and DD. I live over 100 miles away so only see them once a month but for a couple of days once there. Nursery have not commented on his behaviour although aware of it. When I visit always take sweeties and hes not interested just goes into melt down. It's got that way I dread going for a visit as I think Ive got two heads. ?

It may be social anxiety: three and a half is quite young to know all the 'rules' about mixing with others, so I would be wary of labelling it. Children develop at different rates.

Is te walking into nursery with his hands over his eyes connected with you, or just a behavioural thing?

I agree with your thoughts Nanna1 on his being on the autistic spectrum, have to say ( with some experience of this) that it sounds very likely.

Just a thought have you ever taken him out ( without parents) or looked after him when parents go out he may connect you with the parents 'disappearing'
I d certainly wait and see, no point in labelling him at this stage, take him some sweets or little car when you see him and see if that entices him
You don't say if it's always happened or it's a new way of behaving

Speaking of the parents, are there any tensions between you and them? If there are, he might sense it and be reacting to that.

Kids go through some weird stages where they suddenly "don't like" certain people for no apparent reason. Maybe you have a different eye color than the rest of the family and that seems strange to him right now. Or maybe he doesn't like the perfume you wear.

Those are just examples. Could be any little thing. And it could change all of a sudden, too, and he'll be all hugs and kisses. If it hasn't been going on long, I'd wait and see. I certainly wouldn't start labeling him.

If it's been going on a long time/several months, that might be more of a worry. What do the parents say?

This is slightly troubling behaviour if it happens every time. Might be worth chatting to HV and nursery about this