I have figured this out now. I didn't realise the UV Peeler only works on a shape that is extruded, not on a shape with only one layer of flat polygons (I extruded the ribbon, applied the UV Peeler and then deleted the extrusion).
However, when I applied my ribbon texture, which is a ribbed-like texture, it does indeed follow the direction of the geometry as I want but is bunched up where the geometry curves, see attached. I need the 'ribbed' pattern to be evenly distributed along the geometry.
Any idea how to correct this?
7_Days_of_Gifting_Model_UV_Test_No3_v2.c4d