If mom and baby are healthy, both places where I work, baby goes on mom's chest right after delivery, and stays there for skin-to-skin contact and bonding/breastfeeding. The baby does not leave mom's breast until Mom is ready. Then we bathe and do the head-to-toe and gestational assessments. The only thing we do do, is make sure the eye meds are given within first hour after birth and serial vital signs during that first hour to two.
Now, we will bathe and weigh baby if Mom wants it done right away----some people simply can't handle post-birth immediate skin-to-skin for whatever reason. It's up to the family what we do, but I encourage skin-to-skin and tell them why.
So yea, I think skin-to-skin at least the first hour is BEST for the new couplet. Babies are shocked enough after birth, and their transition to extra-uterine life is dramatic and can be rough. It's much easier for the poor kid to learn to breathe and adapt to extra-uterine life if we don't rush the baths, meds and weights right away. There is evidence enough to support this; we are in the process of making it (immediate skin-to-skin contact after birth) a written policy in one hospital at which I work.
So, no, don't let them whisk your baby away from you unless there is a medical necessity, for either of you. It's not good or necessary to rush things that first hour. I recommend you make your preferences clear before the baby is born, so your caregivers know how you feel and respect it. The baths can wait and vital signs can be done while you are holding, bonding with, and breastfeeding your baby.
Oh and don't get me started on bottle-feeding breast babies. SURE your baby needed something to eat, 6 freaking hours after birth!!!! That was of their doing, to have this have to happen. That is one reason why taking newborns to the nursery is such a poor idea, right after birth. That first hour after birth is the best time to initiate breastfeeding, when the baby is awake yet quiet and taking it all in. You missed a golden opportunity to get breastfeeding off to the best start, just by their taking your baby away from you so early and for so long.
Just be sure to tell them, unless there is a strong medical need, bottlefeeding will not be done with your baby for any reason whatsoever. (I can't believe parents still have to say this). There is no need to give a baby formula, unless blood sugars are unstable, and then, we give by cup, not bottle. I think it's horrible to bottlefeed a breast baby without prior and informed consent from the parents! Sorry I guess I digressed on the breastfeeding issue. (like I said, I should not have gotten started)....
Best advice? Talk with your doctor, and the nurses about your preferences. They are not unreasonable. And HOPEFULLY, this hospital has since "seen the light" on some of their rather questionable practices and changed policies regarding early bonding and breastfeeding for new couplets.
Congratulations and best wishes for a safe, joyful birth experience for your family.