I work in the U.K. and have a pretty well respected and well paid career and I wear sleeveless tops AND have visible tattoos.
I agree about the tattoos looking tacky. Most of them dress quite inappropriately for work. I find it amazing how it's acceptable work wear for women to wear sleeveless tops in America - I see a lot of US newsreaders wearing those too. It looks really unprofessional. My work has such a strict dress code - tops have to be at least elbow length long, skirts and dresses have to fall below the knee, no visible cleavage, jackets must be worn for meetings with clients, no tattoos on display ...