Lockdown pends for one tick. And it starts pending when the faction's current state has finished. So it isn't necessarily the case that CG starts; lockdown the next day.
To go into a little more detail, because it's useful to understand *which* CGs are likely to hit Lockdowns, and which ones won't.
A lockdown will go Pending once the conditions (a sufficient excess of lockdown-increasing actions over lockdown-decreasing ones) are met. This will happen regardless of what other states are in effect. Once the Lockdown has gone Pending there is no need to carry out lockdown-decreasing actions until it goes active, as they can't stop it happening.
This Pending period will last a minimum of one day, but can last longer, as Lockdown is a 'medium' priority state. If the current faction state is None, Boom or Bust, (possibly also Famine and Outbreak - I can't remember) then the Lockdown will override that state and go into its active state.
Once in its active state, it will then last a minimum of three days *unless* a higher priority state reaches the end of its Pending time. Higher priority states are Expansion, Retreat, War, Civil War and Election. In that case the lockdown will finish early (which can be in zero days - it appears to just vanish from the Pending queue) as the higher state overrides it.
So whether a lockdown starts the day after a CG starts depends on a lot of factors:
1) Is the CG in a popular system or one controlled by a player group? If so, the CG faction is likely spread across multiple systems, and superior states are very likely to be in progress or already pending as the CG starts. The player group may even be able to easily set them up intentionally.
Where the CG time and date is known (or suspected) in advance, setting up superior states in advance is more practical. (More foreshadowing of future CGs in Galnet I think would be good for various reasons)
2) Does the CG itself consist of lockdown-countering activities? A bounty-hunting CG will produce far more anti-lockdown actions than any plausible number of players can counter. A combat zone CG will enforce a War/Civil War state across the factions for the duration (though the losing faction is likely to end up in Lockdown afterwards!)
3) Do CG participants attempt anti-lockdown activities? In a trade CG, submitting to NPC pirate interdictions, destroying them, and handing in the bounties - if multiplied by the number of participants - should be sufficient on its own. Of course, some people use shieldless unarmed traders, so that they get more cargo delivered personally. If everyone does this - or otherwise evades the interdictions to trade faster - then ... tragedy of the commons ... everyone gets a lockdown and trades much much slower.
Certain missions also reduce lockdown - depending on the CG it may be possible for people to incorporate some of those mission types into their return trips. (Surface scan and surface attack missions are the most likely to be available; some economies may have weapon-related cargo missions)