The tfl/Connect service only runs twice per hour and recently has suffered another bout of unreliability. There have been days recently when they've had zero service into the airport. The service can only use one type of train (unique signaling in the airport tunnel), so when they hit a technical problem it isn't possible to swap in an alternative type of train.
I'd use the Piccadilly line. I know it feels like the journey lasts twice as long as it actually does, but I just wouldn't trust Connect at the moment.