I played competitive FPS games for years at high ranks with a Bluetooth headset before learning of the delay. I did not notice. However, for a while I switched to a Bluetooth keyboard and until I realized the input delay was severely impacting my performance. Wireless mice (non-Bluetooth) on the other hand have low enough latency that most pros are fine using them.

Some advice for Bluetooth headset gamers - Bluetooth completely breaks the audio for some games like Battlefield 1. The only solution is to go into device manager and disabling Handsfree Telephony on your device.