With dd1, I didn't move around as much. I pretty much let the nurses tell me what to do - so was wheeled into the nursery to watch her first bath, then wheeled to my room where I hauled myself into bed. I think I could have walked, if I'd tried, I just didn't.
With dd2, we stayed in the delivery room for about an hour after she was born; it took her that long to get interested in nursing. So didn't attempt to walk, just stayed there with her 'til she had a good nursing session.
IMO that's the #1 goal after birth, is to get baby to breast and latched.
I believe I just walked to my room after that, no problems -- the nurse helped me clean up a bit in the bathroom, and then I settled back in bed with dd2 and snuggled/nursed. DH went and got me dinner because I went into labor during dinner time and hadn't eaten supper, but they were done by the time I was done nursing etc.
I was definitely much more mobile more quickly after dd2 than dd1; both were unmedicated births. But with dd1, I hadn't been able to sleep at all the night before she was born, and worked all day that day at work -- then locked myself out of the house on a cold October day and had to wait two hours for dh to get home (he was driving as fast as he could). Then back labor with a posterior baby, so I was really tired after her birth - much more so than after my quick labor with #2, after a day spent 'nesting' at home getting projects done and an early-evening delivery.
With both, I was definitely mobile and moving around easily the next day -- my only problem has been a very sore tailbone after both. I suspect that has to do with the near-upright (but still in a bed) delivery mode with both girls. If we have another, I'm going to tell them I will be delivering in a squat or on my hands and knees, either of which should I hope take that pressure off my tailbone.
I would say that length of labor and amount of food/water withholding during the labor have a lot to do with whether a mom is mobile afterwards, with a natural labor as well as a medicated one.