They each have a range of other meanings as well, but in your example they're interchangeable.

You can use any of them. "Sorry to bother you" and "sorry to disturb you" mean you are sorry to be annoying them. These are polite because they recognize their feelings. "Sorry to interrupt you" means you are sorry to be breaking into their conversation. It is less courteous than the other two choices because it doesn't recognize their possibly hurt feelings.

To interrupt= to stop someone from speaking by saying or doing something. To bother = To annoy, worry or cause problems to someone Exemple: Don't bother your father when he is working. Disturb= to cause someone to stop what the person is doing. Exemple: Please don't disturb Jimmy - he's trying to do his homework. "Interrupting" and "disturbing" someone isn't necessarily annoying which isn't the case with "bothering". In this phrase only "interrupt" and "disturb" are interchangeable, however not "bother".