Just would like to mention, maybe test teatree on a small area first (put a little circle of teatree oil diluted in whatever carrier oil you use, on inside of baby's arm, wait 20 minutes or so, then check for signs of irritation/welts/redness, etc. If the teatree/carrier oil dabbed skin deviates from normal looking in any way, do not use.
Just to make sure your baby has no allergy to the teatree oil. I found out I was allergic to teatree this way and I'm very, VERY glad I skin tested before applying a whole lot of it.
Teatree is a great antiseptic but it is also strong. The active chemical in it is like a terpene, which makes it somewhat chemically similar to turpentine.
I now test all herbal oils/lotions/shampoos etc. with the skin test before using them.
Hope your baby feels better soon.