I can see this from both angles, having been infertile and going through 10 years of investigations, some painful and unpleasant, in several countries. I was incredibly lucky in the end to have a baby by IVF, and then another, again by IVF, three years later. With my first I had an emergency caesarean, followed by my baby having to be kept in Special Care for several days as I'm a Type 1 diabetic (this doesn't happen now thank goodness!). So when she finally was allowed to stay in my room with me, I was incredibly nervous of anyone holding her, even myself! I know all the in-law family held her, as I have photographs, but all I wanted was to cuddle her, get to know her, breast feed etc.
My MIL and SIL were very critical, once my baby was home, about my feeding her on demand, cuddling her when she cried etc. etc., but my memory of my mum is hearing her washing up in the kitchen while I spent time with my daughter, and no criticism!
When I became a granny, twenty-two years later, I didn't meet my granddaughter until she was about eight weeks old, of course I longed to hold her just like the OP does; surely that wish is totally natural? But I was led by my daughter, holding the baby while sitting, watched by her, and handing her back immediately when asked to. I have always respected hers and her partner's way of parenting and just tried to be supportive and helpful.
OP I can see that your wishes to hold the baby (I think it is instinctive in women) are completely normal and natural. I can feel your DIL's instinct to protect her (hard-won) precious baby from any risk at all, and even to be a little more protective than other new mothers; completely normal and natural too.
Neither of you deserve criticism for how you feel; just help to see why the other may be behaving as they are. At this moment in your grandchild's life, his parents, especially his mother, don't have time or headspace to do anything more than to be new parents, so please just lovingly give your DIL space to do this. If she feels supported by you, your future relationship with her, your son and grandchildren has a chance of developing into a warm one.
I can visualise you both sitting down in the future, and chatting about the time when you were SO desperate to hold him, and she was So desperate to protect him from the tiniest risk!