I would just make sure to
bathe the wound in betadine and dab some neosporin on it. You can apply a little pressure and some corn starch to stop the bleeding. That works on cats when you cut their veins in their claws as well and is recommended by my vet. I would also make sure your beardie has a clean cage floor where they can't get any debris in the cut while it is healing.
And if you have those screen doors, I would remove them. One of our oldest, Mike, had snout rub as a rescued juvie from his doors and our vets don't recommend screen doors with beardies because they can get injured easily.
I realize these answers might be a little late to help with everything, but I hope some of it helps and you know for future reference now what to do.