You could consider discussing the following rule with your daughter: 'I will come and stay but only to undertake shopping, cooking and cleaning. I will not interact with the baby unless asked'.
This worked wonderfully well for myself and daughters: the new parents get to know their baby and settle into their own routine.
I was asked to help - very quickly, too! Matters such as the best angle for breast feeding or best way to stop the baby crying (swaddling helped), were put to me and when they worked, everyone relaxed a little more.
Having a baby is a terrifying time, especially now. I vividly remember arriving home from hospital, my husband plonking the car seat down and us looking at the sleeping baby and wondering 'what are we supposed to do with it now?! Perhaps your daughter is concerned about your having Covid? You could reassure her that you would take a lateral flow test daily, or similar?
Whatever you do, do not flout your daughter's wishes as it will come back and bite you. She makes the decisions.
As to who sees the baby first, well, the baby won't have a clue and does it really matter? Yes, of course you will be a little hurt but distance is a logistical problem for you and you will have to accept that if local, the 'other' grandparents will see more of the baby. You can still have a full relationship by Zoom, or in person so please don't fret too much. There is about to be a wonderful new addition to your family, try to relax and enjoy it!