maybe you already solved the problem yourself as this thread is now two days old. I think in Publisher everything works fine. The Problem is with the imported Affinity Photo file. After some playing around with several solutions for transparencies in Affinity Photo I could find out wich ones would work for text wrap in Publisher. But first the possibilities that won't work
What doesn't work is:
New File > do not check "background transparent" > create new file > draw a path or shape > rasterize it > export as PNG, GIF, TIFF or safe as .afphoto.
Open file > use normal layer masks for cutting out objects > save as PNG, GIF, TIFF (without layers) should work but not with an .afphoto file...
What works is (allways set background to transparent when creating a new file):
New File > create a shape or vector path > rasterize the layer with the object > export as PNG, GIF, TIFF or save as .afphoto
New File > create a shape or vector path > export as PNG, GIF, TIFF or save as .afphoto
Open File > use the "Pen Tool" to draw a vector path around the object you want to cut out > click "Mask" in the left corner of the context toolbar > safe as .afphoto
Open File > use any selection too > create layer mask > "Rasterize", "Merge Down" or "Merge Visible" layer(s) > safe as afphoto
I think for you could work the 4th solution. Text wrap works with (vector) shapes and vector paths, vector masks, raster images with true transparencies like GIF, TIFF (without layers), PNG and .afphoto files where masked layers are rasterized. Any mask layer active in the saved .afphoto file will interfere with the text wrap styles "Tight", "Inside" and "Edge". I Hope this will help.
New Year's greetings