In this circuit the first OP-amp needs to be quite fast to overcome the read zone of about 2 diode drops. So the slew rate can be critical. The slew rate of FET OP-amps are measured at a rather high input differential - so you won't see the advertised 900 V/µs.
There a similar circuits that lower (e.g. reduce to about 0.2 V) the dead zone somewhat. Another point may be the transistor - there may be some delay in turning off. The more typical circuit just uses a diode, and needs a fast diode, like a small schottky diode. Also the capacitor may have to be rather small (e.g. 10 pF range). If needed a second similar (bus slower) stage can be used to cope with drift due to the small capacitor.
The alternative is using a simple diode - capacitor circuit without OP-Amps for feedback, but with compensation for temperature drift and maybe nonlinearity. At low frequency this is less accurate, but it's much faster.