It varies.
Sometimes baby needs resussitating in which case it'll be brought straight over to the resuss table and worked on.
More usually though that's not necessary and in most hospitals they'll place baby straight on your chest and encourage you to latch on. The cord is clamped before it's cut, depending on whether you have the injection or not this could be anything from seconds to hours after the birth. Usually if only minor suction is needed they'll do it in your arms or else they'll bring baby to the resuss table to do it.
It all depends on the type of bith and the hospital in question, for my three:
DD was born blue... they let me hold her for a split second before she was whisked away to be resussitated, i got another quick cuddle and then she went in the incubator as she was cold and wheezing. They stitched me up with her in the incubator beside me and i had my hand through the hole touching her and holding her hand most of the time she was in there.
ds1, same hospital but better birth, was a physiological third stage so i got to hold him and feed him while we waited for the placenta, then he was weighed. tagged and dressed and given back to me. I held him while they examined me and made sure i didn't need stitches.
ds2, different hospital, physiologial third stage again so i got to hold and cuddle while we waited for the placenta again, then he was weighed, tagged and dressed and as soon as i'd been checked we went to the ward. Unfortunately he had aspirated pneumonia so we weren't there long when he was admitted to SCBU.