I see. In that case I’d use a MoSpline referencing the spline that’s connecting your two objects. You can sweep the MoSpline and you can also animate the length of the MoSpline along the reference spline. If you go that way you’ll have to keep in mind that the MoSpline’s Spline>Width parameter controls the start radius of the sweep, as opposed to the radius of the profile spline that’s being swept.
Unless you’re using the Hair post effect or Sketch & Toon, you’re going to have to generate some geometry to render that line. So use a Sweep Nurbs object and animate the growth parameter.
It seems that you have to have a Texture View window open in order to activate the UV Poly tool, at least initially. You could create a layout with a tabbed Texture window if you wanted.