Would just like to report back that we ended up taking a taxi.
Our flight was well over an hour late into MAD & the only way we were going to make it to Atocha was Taxi/Uber. I ruled out the latter when I saw the queue for the lift to the pick up point. We dashed back to the Airport Taxi queue, which was rather long itself. We arrived at Atocha with only about 10 mins to spare.