You can speccify - to some degree, all birth plans have to be flexible - what happens. I wanted the baby to be delivered straight onto my chest. Dh knew how strongly I felt about this, and immediately the baby was fully out dh was pulling off my remaining clothing so that the midwife could lift the baby straight onto my chest. OMG, the most amazing sensation . But not everybody wants that. If you want to specify that the baby should be wrapped in a towel first, or wiped clean, you can.
Unless the baby is ill they shouldn't take him or her away from you at all - again, not unless you request. They will want to weigh and do some medical checks, but mostly these don't need to be done immediately. I think it's things like chest clear of mucus, breathing well, skin getting pinker, temperature, response to stimuli. All very quick checks. Many of them can be done without taking the baby away from you. You may not even notice the mw making her observations. Your birth partner will be able to have a cuddle as well. As long as you and the baby are well it's entirely up to you.
They may wash the baby for you, or offer to help your partner do it, or you don't need to have the baby washed at all. It's healthier for the baby not to wash, but it also depends on your tolerance for gunk .
After a while you'll need to be checked to see whether you need stitches. My dh held the baby for this bit, and also for the stitching itself. Once that is over you'll probably want a wash. What happens then depends on the facility. Dh and the baby came with me each time.
And then you get to go up to the ward - though, depending on how busy they are, you might be discharged direct from the labour ward if you are OK. I think that I went to the PN ward about 2h after the actual birth in each case.
TBH I enjoyed the wheelchair ride to the PN ward - the first time I needed it as I'd had an epidural, the other two times I requested it as I felt I'd worked hard enough to deserve the ride . Also I got to carry the baby, which I wouldn't havbe been allowed to do had I been walking - he would have had to be wheeled in his bassinet.
HTH