I often itch from one of the cats' biting, although rarely do they break the skin. The only time I've ever gotten ill from a cat was when Esse nailed me badly near my eye...but that was significant infection, and happened really fast. I was surprised at how fast I got ill, too. Something like 8-10 hours and I needed to have the bite lanced (really disgusting and smelly), disinfected, and massive doses of antibiotics. Let me stress again that it happened fast. Not slow, but really quickly. Overnight.
One thing I've noticed in my life is that blood itches. If I cut myself, but don't wash the blood off (skydiving accident, or rose bush attacks, or a rambunctious kitty getting a tad too rambunctious in their play with me...something like that...) I've noticed that if the blood is left to dry there, or rubbed away rather than washing up, it can become itchy. Not a bad itch, but still, annoying. And then even after I wash, it stays irritated. Dunno why, I've got clean blood and all, but still...it's what happens with me.
Try washing the whole area very well, with an antibacterial soap if you have some. If not, use some vinegar, which kills most surface bacteria. Apply some neo-sporin, if you've got it, or some antibacterial cream. If you haven't got any of that, try some aloe. Then stick a bandaid on it, so you remember to not itch it, and watch for infection sypmtoms (although I don't think you've got any, or it would've likely showed up by now...). If there are some, then get thee to a Dr. right away, to heck with Christmas.