Repair service specifically for bike commuters?
You don't need such a repair service.  There is a different, more practical solution: a loan bike.

In The Netherlands, most bicycles are used by commuters, so expediting wouldn't work.  However, there are also many places that rent bicycles.  A good bike shop can simply have a couple of dozen bicycles on hand that they then loan out to a (regular) customer, so that they can still go to work, school, social outings, etc..

This is already common practice in The Netherlands.  I believe there exist car garages that do the same, so I don't see why it would be any different for bicycles.

If the bike shop is unwilling to loan you a bike, and you have the money, you can just rent a bike to cover the period you won't have access to your own.  If it happens frequently, you can buy a secondary bicycle for the same purpose.