My approach to dealing with a toddler who wants to nurse ALL THE TIME (more than the baby) is to borrow from weaning techniques and distract/offer substitutions as much as possible. (I'm mainly doing this because DD1's latch is AWFUL but that's another thread.) Reading books while nursing the newborn often proves a good substitution. LLL suggests having a basket of specific "while baby is nursing" toys.
Also, this sounds crazy, but I've found that it's easier to take them both out of the house, mainly to places that DD1 views as "fun" (grocery store, library, restaurants, etc.), than it is to be home alone with them. DD1 is less likely to demand to nurse, the dogs and our breakable glassware are all safe, etc.
A baby carrier that provides the newborn with good head support makes it easier to deal with the toddler. I like a mai tai, but there are lots of other options that would be just as effective.
And this goes against all my parenting values, but dvds are a cheap, effective babysitter...
My husband is gone for 4 days at a time when he works. I got through his first "weekend" back by having my parents come up to entertain DD1 during the day, but I still got to figure out bedtimes and showers on my own. The next time he works DD2 will be over a month old and I'll be totally on my own. I DID manage to get them both to an 8 a.m. well baby visit all by myself the other day... it wasn't as difficult as I thought it would be; the biggest challenge was getting myself out of bed.
Oh if anyone is trying to figure out showering while home alone with a baby, an infant seat in the bathroom is your new best friend. If you have a toddler too, the best option I've found is waiting until s/he is asleep... although I did live dangerously the other morning and trusted that a dvd would keep DD1 immobilized in front of the computer screen while I took a very quick shower; it worked, but I'm not sure I'd try that with a kid much younger than her and it was pretty nerve wracking.