Hi,
How fast are you logging the data?
The Crazyflie state estimator, a Kalman filter, is running at 100Hz and does not exhibit a lot of delay: if there was any significant delay in the state estimation the Crazyflie would not be able to fly autonomously. The delay might be caused by a slow log or packet buffered somewhere.
How are you moving the Crazyflie? Are you moving it manually or is the Crazyflie flying by itself. The Kalman filter has two different behaviour for these two modes.