You've gotten a lot of good responses already. But I'll add my 2 cents. I've nursed 3 babies and not done any bottles (Mine are nearly 5, 3, and 16 months). With my first I waited way too long to leave him, but by my 3rd I was much more confident. I found that I could "tank" the baby and then be able to leave him at home (for short periods) with my dh. I also found that as they get just a little bit older they start going down for their first long stretch of the evening at a somewhat consistant time. Consistant enough for me to plan a late dinner out with a friend and leave baby at home. Alternatively, I could take the baby with me and just go to Walmart and have baby stay asleep in the carrier. That way I could nurse (in the car if need be) and not have to worry.
As the baby becomes a better nursers. NIP may be come easier. They don't need as much help - with the latch and with head support and all. (of course, then it gets hard again as they become preoccupied with the world beyond the blanket). With my first I bought a lot of nursing tops -- they provided me with the confidence I needed. Now, I try to wear a tank with a button down over it. That way I can lift the tank and have the button down cover my side. I've always felt way more exposed showing the world my stomach and side as I nursed then the .2 seconds my boob was exposed as I latched the baby on. I 2nd the vote for burp rag/ cloth diaper as a cover up on top for NIP and it's nice in the warmer months so that baby isn't getting too hot under a blanket.
Finally don't underestimate the power of a walk around the block. Sometimes just the fresh air really helps. And baby can survive 15 minutes without you.