These are the commands I used:
/skin set URL classic "http://axelmod.great-site.net/minecraft/fluttershy.png"
/skin set URL classic http://axelmod.great-site.net/minecraft/fluttershy.png
/skin set URL slim "http://axelmod.great-site.net/minecraft/fluttershy.png"
/skin set URL slim http://axelmod.great-site.net/minecraft/fluttershy.png

I got this return:
Trying to set the skin ... Please wait.
This player doesn't seem to have any skins saved.

And nothing happens.
Of couse, I also tried to use different skins:
/skin set URL classic "http://axelmod.great-site.net/minecraft/twilight1.png"
/skin set URL classic "http://axelmod.great-site.net/minecraft/twilight2.png"
/skin set URL classic "http://axelmod.great-site.net/minecraft/sailormoon.png"

All of them, with and without quotes,
all of them either classic and slim.
And nothing worked.

I also tried to upload them from my local drive and it kept not working until I was writing this.

I did not understand what was the problem. Do you mean the problem is my web server?
Yesterday I installed XAMPP on my PC, so I do have a new server to make the same tests, but maybe it could be done on this weekend. I am too busy this week.

It's nothing I can do about, FabricTailor uses mineskin api and you cannot even upload skin from that url there. Try contacting them instead.

This player doesn't seem to have any skins saved.

Huh, seems like wrong method is getting called ...

Just checked on mineskin.org - it's a problem on their end