I picked up a used band-aid at work, what are my chances of contracting diseases?

About zero. Having said that, what was the bandage like and where do you work? Was it covered with pus and did you have a break in your skin? Next time use a glove or at least a paper towel to act as a barrier. You have a personal responsibility to use your head; your employer can't babysit every thing you do on the job. Use common sense!
Miniscule. Extremely rare unless you had/have a huge gaping wound and the bandaid was dripping blood.