How do you think you build a good, safe and dedicated bicycle infrastructure in a city which has not been designed for it?
The Netherlands did it. Just change construction requirements/guidelines, zoning, etc, get some biking activists, and wait 50 years. All of these problems have already been solved.
And to answer your specific question, I think they normally close off roads to regular cars but let delivery vehicles go through. In the short amount of time the vehicle is there, people just bike/walk around it. And they also make smaller delivery vehicles, including branded cargo bikes for when the situation fits.