DD runs to the sitting room to get her stocking and Santa presents as soon as she wakes (well, usually after thundering through our room to say she's going down).
If we are at home, the rest of the presents are opened in the late afternoon - after we've been to mass, done the few visits to relatives living near us, and got the turkey in the oven and the stove lit and the drinks open (and nibbles cooked).
If we are visiting DPs, DPILs, we get a pile thrust at us as we arrive into the kitchen at DPILs (whether that's from a bedroom at bleary-eyed o'clock, or on arrival after mass in late morning). When we then get to DPs in the late afternoon, they are just about to or not long started the process there.
In both our own house and my DPs, someone hands out the presents individually to be opened, admired, thanks issued and then the next one gets handed out.
Last year, at ILs, presents were done after lunch in DBILs (next door) as they cooked and hosted. Each person got a bag or bundle of their gifts, and starting with the youngest, 1 person at a time opened everything they got, before moving on to the next oldest.
So every house has a different way of doing it!