It does and a lot.
CWA is a constant current, so when the lamp is still cold and its arc voltage low, the power dissipated in the lamp, so the heat warming it up, becomes low as well.
On the other hand the series reactor and to some extend also HX autotransformer tend to operate very close to saturation during normal operation (to minimize the number of turns needed, so the wire length and so losses).
This has the consequence of becoming partially saturated once the arc voltage is still low. And a saturated choke (or shunt in CWA) means the current becomes significantly higher than would otherwise correspond to the reactance it has at normal operation.
And this higher current at low arc voltages means higher power dissipated in the lamp, so more heat warming the thing up, hence the shorter warmup time.