It does vary widely but just to mention that your baby is at risk of possible blood sugar issues in the first few days after birth since you have GD. So there’s a greater chance your baby may need some extra medical care or perhaps even a short stay in special care. So please don’t get too set on exactly what you might want to happen as birth is always unpredictable and some unexpected things might need to happen.
Having said that, generally if both you and the baby are doing well, you will hold / cuddle the baby, skin to skin if you like, for a period. The newborn checks on a healthy vigorous infant don’t take very long and I was asked whether i wanted my husband to dress the baby for the first time. Generally baths are not advised right away as babies can have trouble regulating their temperature and as the vernix may have benefits for their skin.
If you can’t shower right away because of weakness or other reason, the midwife will give you a sponge bath, which I found way less undignified than I’d imagined it to be. When you move will depend on a whole host of things like demands on delivery suite, etc so that could be anything from an hour or so after birth to a longer stretch.
The baby will be weighed and have ID tags and security bracelets placed on her. Your baby should also get the vitamin K injection as part of the newborn check but it is really no big deal at all, and is over in no time.
For you, you may be asked to measure your first urine after birth if there are any concerns about potential injury to your bladder - this shocked me a little!
There are loads of variables though and in general everything will be explained to you and there are no major choices to be made, so you can just go with the flow if you like.
I find an unrealistic emphasis on many blogs/advice sites about uninterrupted skin-to-skin etc being absolutely necessary which I find unrealistic in setting expectations.
Also be prepared for any one of an enormous range of emotions. You might be exhausted and feel a bit numb. You might feel shocked or traumatized at the enormity of what has happened. You might feel detached and uninterested in the baby. Or you might feel the Hollywood moment of an instant fish of love. All are entirely normal and again, don’t feel any pressure about what you “should” feel.
Best wishes!