Is it effective to wash your hands with a Fairey type of detergent if you run out of soap?

Effective, only sorry for the hands. I wouldn't wash the faerie dishes. The composition is terrible. Use mustard powder, baking soda, and a bar of the cheapest soap.

Effective? Yes. Washes away oil, gasoline, paint, etc. Helpful? No, but there isn't much harm from it either. Fairey works on the basis of sodium laureth sulfate, the same surfactant is included in many products for washing hands and hair. But here you need to monitor the individual reaction. Perhaps after washing the skin will dry very much. Better not to use it all the time (or use a moisturizer after washing).