"My uncle and my father were drinking fruit juice that day."
Translation:Min onkel og min far var ved at drikke frugtsaft den dag.
You're right that "ved at" can also mean that you're in the midst of something, but as a native speaker, this translation still sounds wrong. If this was in present tense they could be in the midst of drinking, but in past tense the only acceptable translation is that they were about to drink.
English has always been able to be translated both ways. "They drink" and "they are drinking" means the same, just like "they were drinking" means exactly the same as "they drank".