It depends on the cooldown. Safety cooldowns are completely unaffected and still take their 30 or so seconds of reboot.
But if you jump from one system to another, the FSD seems to get briefly deactivated after the completion of the jump. And the following boot-up time is shortened.
If this subject is of interest to you , perhaps try a few tests yourself to see what happens.
I did investigate this and I've observed the difference between an FSD with faster boot sequence and one without many times. In addition to the combat ship scenario, you also have a situation where you disable the FSD on an exploration ship.
If you do about 20 neutron star supercharged jumps, your FSD health will have dropped to about 80% where it starts to malfunction. At this point you would drop out of supercruise, enable your AFMU, and then repair the FSD which turns the power off to it while repairs happen. When you active the FSD again, that is when the boot sequence happens. On my Phantom with the Engineered FSD V1 it boots in a few seconds - like 2 or 3. When I once went exploring in an Anaconda with a regular FSD with increased range engineering (so no faster boot sequence), that boot sequence time after repair was maybe 4 times longer. This is what the Faster Boot Sequence modification does, in addition to a small improvement in jump range. There is no other difference.
Safety cool downs during which you cannot jump or enter supercruise are unaffected by the Faster Boot Sequence modification. But all safety cooldowns are not the same. If you try and fail to win an interdiction, or fly into the exclusion zone of a non-landable planet, star or blackhole, the FSD does an emergency shutdown and requires a long cool down period - I think that's your 30 seconds. If you complete a normal jump, there is a controlled safety cool down period of around 10 seconds before you can charge the FSD to jump again. If in supercruise, you slow to minimum speed or arrive at a navigation point and drop out of supercruise, you also have the controlled safety cool down period of about 10 seconds to wait before you can charge the FSD again.
If in doubt, run some tests yourself with the different ships you have.