Ha - you sound like a stay at home mom.

You might do better in a day care setting with other adults around

Take a class or join a book club after hours. You can try to find a regular a story time and park to meet other nannies and parents who are chatty and open to meeting up again. This site will give you some laughs from time to time as well. A job like you described is a bit isolating, you should by nature be ok with that. Oh, hand work like knitting, embroidery, etc. will also make the time pass.

Or, you may not be a good fit to be a nanny. If you need a lot of socialization during the day, being a nanny may not be the right job for you.
Yes, you can have play dates and meet other nannies, but there will always be the time alone with the child. And if you can't handle that, maybe look for something g that will offer you more socialization like a preschool or daycare setting, or find a job with another sahm. Too many nannies complain about having to deal with a sahm, but it could be what will work for you.

I say get out more and involve yourself in activities that involve more parents. And when there are other parents involved, take some initiative and start conversations w/them.

If you have good chemistry w/them and they live close to where you work at and your schedules are similar, perhaps arrange a play date or even a lunch date for everyone.

Also, during the day to avoid the ominous silence of the house, why not play some children's CD's?

Take walks, find free events at bookstores/libraries, get MBs approval to meet up with a friend who is great with kids for lunch, call a sibling/friend and chat during nap time (of course after all chores are done...), listen to music or podcasts when you are stuck in the house, cook awesome lunches for your charge (that take more than 10 minutes) and have them help as much as possible, find age appropriate crafts, make friends in the neighborhood...that's what I've got for now. Get creative OP!

Playdates

How do other nannies avoid this? I recently left my job working with a stay at home mom and her two kids (3 and 6). I had company all day, stayed quite busy, and loved it. Now I'm watching a two year old with delayed verbal skills and two parents who work 50 hours a week and I'm going a bit crazy. I talk all day and narrate our activities to my charge, but am going crazy with nobody to converse with. We go to the park, storytime, music class, but there is still so much time during the week that I feel utterly alone.