You can get out and about as soon as YOU want to. I dont think that can be predictes in advance though.
DC1 - popping out to the shops etc on day 3 (discharged from hospital early day 2). Did some socialising on day 5 and had visitors on day 6/7. Had seen my parents day 2 onwards at their house or mine. However, socialising out of the house was brief. I had a small tear, and even though it dodjt need stitches, having a wee STUNG. So I wouldn't have gone out for lunch as I wouldn't have wanted to wee! At 2 weeks I drove 4 hours to attend a friends 30th party. Great fun. You can dance with a sleeping newborn on your chest, the music doesn't bither them at all! You can't eat/drink without sitting at a table though, not enough hands (didn't take sling). Had a fabulous night.
DTwins were lockdown babies, so nowhere to go. But I wouldn't have been up to doing much for at least 2 weeks, probably closer to a month. They were 6 weeks premature and needed to be in incubators for a while. I was expressing milk every 2 hours, and juggling looking after eldest (no supportother than DH the to lockdown and nochildcare) and visiting twins to try and feed them/do skin to skin. When they did come home I was exhausted. I found the recovery from the emrgency section pretty tough too. So no way would I have been up to socialising.
DC4 was a very easy birth. Discharged within hours, felt pretty normal and comfortable. Not even very tired as she was sleepy so I basically slept at the hospital until ready for discharge. Lovely! I was at a toddler group the next day running after my twins, newborn in the sling. Did school run from day 3 and had no issue doing the activities my eldest 3 did.
So I'm my experience it really depends on how the birth goes. You can't totally plan for that. So best to have no expectations and just roll with it.