Will the wrinkle under my nose go away from the lip filler? It's been a year or is my Botox causing it?

It is difficult to tell without a photo which type of wrinkle you are referring to.  It sounds like a wrinkle from aging/sun damage but if you post a photo it would be easier to answer.  These type of wrinkles are best minimized through Fraxel and facial fillers such as Restylane.

I notice that some women develop a line under the nose above their lip as they smile. This is unlikely related to filler from a year ago.   Botox may help.

If by line you mean the transverse line, this can be corrected by filler injection, laser, or a combination of both. It is unlikely cause by botox or filler though.