It happens on initiative count 20. It renders him totally immune to all spellcasting and magical abilities. In addition it renders magical weapons non-magical.... which the Lich is immune to.
It is debatable whether or not divine smite is magical in nature. Seeing as it requires the expenditure of a spell slot to function, I would rule it is that it is magical and does not function in an anti-magic field. I'm also assuming the Lich is hovering from between 15 and 20 feet in the air, out of range of anyone trying to grab him or similar shenanigans.
From there the Lich is almost certainly immune to anything the party can throw at him, has his turn, and 3 legendary actions.
Howl, life drain, flight, and Energy drain can all be used in an anti-magic field. I would have it howl on its turn, then use energy drain and fly up to the roof out of reach of any melee combatants.
On turn two I would use the lair action knocking everyone prone, followed by the trap soul ability, followed by the cloud of dust and energy drain legendary actions.
I would then on turn three re-established the anti-magic field as my lair action, and use my life drain ability to heal any hit points I lost in the previous turn.
You did a nice job planning his actions. But completly forgot what lvl 13-15 characters can do.
The perfect scenario he get a surprise round agaist everyone, use howl and save the antimagic field for the first non surprising round. This is an 75k xp encounter, 3x deadly for 4 lvl 15 players. But this one can really be deadly.
But, if he don't get surprise round, he uses howl, is a Con ST more than half (or more) of the classes at that lvl can bump his saves of some way. Maybe one of four will fail.
Players can't use spells agaist acererak, but can use in his ally, recover the 0hp, or the frighteneds if they're still afgected for this. use buff spells, one round to use any spell they need for buff or recover.
Is really easy for a lvl 14 reach 20ft with magic, harder without, but not that hard, unless they lack total criativity.
Easier ways: teleport above AMF and grab him while you fall. Or climb (buff with haste +action surge or bonus action dash or normal climb speed?)
There is abou 40% of chance of someone act before lair action in initiative too.
And they still kill acererak in one nova round after antimagic field is gone.
Or the players may stay there doing nothing like giant tofus while acererak get them souls.
By what I see in my tablr, better I have a plan B and C.