Hello, my name is***** and I have over 20 years of experience as a veterinarian.
I am so sorry to hear that your dog was exposed to tea tree oil. His symptoms do sound like a toxic reaction as the terpenes in the oil cause central nervous system depression, leading to weakness, muscle tremors, incoordination and mental dullness and depression. In some cases it can be depressing enough to the central nervous system to lead to death.
These are fat soluble compounds and they need degreasing agents to be effectively removed. I recommend bathing him with Dawn dishwashing soap and tepid water several times to remove as much of the toxins as possible. Perhaps a friend of family member can help you do this.
There is no antidote to tea tree oil toxicity, we simply need to remove as much of the toxin as possible and then rely on supportive care while the toxins wear off. If you want to administer milk thistle (active ingredient silymarin) that can support liver function, decrease inflammation and treat oxidative damage. I would dose it at 1mg per pound of body weight per day.
This is a slow to degrade toxin, so side effects they can linger for days, and in some instances with large toxic exposure weeks. As long as we don't have severe signs of toxicity though, and the toxin is removed, most dogs eventually come around.
I recommend that you watch him closely today after bathing him. If she becomes more depressed and his symptoms are worsening even with removing the toxin such that he cannot eat, drink or eliminate and becomes mentally dull then he should be admitted to a veterinary emergency hospital for intravenous fluids to flush out the toxin, warming and support. I understand that would be difficult but that will give him the best chance of making it through this toxin exposure.
Best of luck with your fellow, please let me know if you have any further questions.