Does it reallty matter which g.parent sees the baby first? Hopefully, you will all have very many years to see this baby grown into a child and into an adult and have your own relationship with them. In what way does it matter who sees it in which order in those first few days/weeks.
Back when I had my first child, (ten days after birth in maternity hospital for first-time Mums in those day), I was delighted when my Mum said she would come over to me each day for the first few weeks. By the third day, I was really desperate for her not to come. I was exhausted and instead of having a rest when baby was sleeping felt I had to sit and talk to her. I asked her not to come so often and it was so much better.
When my eldest daughter had a baby and she lived very close to me, I was there when she was brought home from hospital by her partner on the day she gave birth. Stayed a short time, and then left. Knew I was at the end of the 'phone if she needed me, but felt it was important to let the three of them get on with it.
Apart from one desperate, tearful call from her after a very stupid midwive had told her to stop breastfeeding - (she had also been told to take baby to GP as it was not putting on enough weight). I took her to GP, who was wonderful and she commented how lovely and healthy baby was and urged daughter to continue bfeeding.
When I did visit them I would do some ironing, clean bathrooms, kitchen, etc. Sometimes just drop in with a meal that just needed to warmed up. NEVER to try to take over any sort of babycare - unless I was actually asked to do something along those lines.
Lots of ideas on good baby and childcare changes quite often. Many things we may have done back in our day are not likely to be the same, not necessarily old fashioned, just different. So, unless actually asked, NEVER offer advice.
I had, for many years, been a breast feeding Councillor, and my daughters had all grown up hearing me giving help and advice on the phone to Mums. They were willing to ask my advice on that matter in the early weeks, but on everything else they and their partners sorted it out along with their own support groups.