I use Airportr all the time, always from central London to either Heathrow or Gatwick. They send a driver who sometimes has just my bags to pick up, sometimes has two or three other customers to pick up from, but they don't tend to have a lot of jobs per driver, at least not from what I've heard from my drivers. The driver always knows what time he needs to get my bags to the airport and I follow them using my Airtags sometimes, and 9 times out of 10 the bags go straight from my flat to the airport. So although I can't speak for ZRH I imagine it will be the same model, they will have it worked out how to get your bag there on time.

Something to bear in mind is to have a Plan B in case you can't check in online. If BA's IT is as useless as normal, it may not allow you to check in online and you must have a Plan B to get to the airport as Airportr won't take your bags if you haven't checked in. Their customer service is absolutely excellent, they respond quickly to live chats and their staff really try to help if there is an issue. Definitely recommend.