I used to work for Parcel Force, trust me, it would get lost!
Having flown Air Taxi/Ambulance for a few years, I must say that in my experience, as long as your status as a hospital flight was specified on the flight plan, ATC (all over Western Europe at least) has always bent over backwards to ensure direct routings as often as possible. If they didn't immediately offer such routings, we only had to ask.