These clocks depend dilute gasses of atoms. Usually the process for preparing and probing the atoms takes at least a few hundred milliseconds, and usually in the seconds range. There's a bunch of reasons for this, probably two main reasons IMO:

1. Generally for precision clocks you need "cold" or "ultracold" atoms (essentially random doppler shifts of hot atoms kill your accuracy). This means something like microkelvins - this is hard to prepare quickly. And atoms are constantly lost/heated due to collisions with stray gas molecules, even though the pressure is usually something like a quadrillion times lower than atmosphere. There's research into continuous ultracold atom sources, and you could prepare way more than you need and siphon a few off at a time, but both are technically challenging.

2. The way these clocks work is essentially you have some laser, and an atom that only reacts to laser light of a very specific frequency. If the laser frequency is off from the exact frequency you can tell. A variety of effects means that the measurement process takes some time. It's like trying to accurately measure your heart rate in 1s vs 10s - a lot easier to do in the latter case.

Nothing fundamental, just technically quite painful.