Don't feel bad! It's not your fault. The only time it's preventable is when nails get too long and they just get pulled off while running around, but here you were clipping her nails and wham, off it goes as she's squirming out. That happens! Don't worry, I didn't mean to panic you - it'll be okay. just keep an eye on it.
I am not knowledgable about tea tree oil used on bunnies (another thing to put on my list of things to ask about)
Some people rub a THIN layer of neosporin on and hold their bunny for a few minutes to help it soak in. It will leave a bit of a residue as that is just the texture. A thin layer of it should not do any harm, but if she licks it off, it sure won't do any good either to help stop infection. So you can just watch and see what she does. You don't have to put the neosporin forever, but just for a few days to a week (once a day)
ADDED LATER: I wanted to also let you know that neosporin has been recommended for bunny care kits - Example http://www.mnhouserabbit.org/care/f...html (scroll down) as well as in the emergency kit recommendation by the Ontario Rabbit Education Organization http://www.ontariorabbits.org/healt...hinfo7.htm
And the House Rabbit Society in Miami says this in regards to an irritated area: Apply a soothing balm, such as Calendula (from the Health Food Store) or triple antibiotic ointment (e.g., Neosporin, but NOT Neosporin Plus, which contains lidocaine and is not recommended). Source: http://www.bio.miami.edu/hare/buttbath.html
I've used just the regular neosporin on Rucy and Bailey.