The Animate Objects spell description says:
You decide what action the creature will take and where it will move during its next turn, or you can issue a general command, such as to guard a particular chamber or corridor.
This makes it clear that the animated objects have their own turn, but it doesn't explicitly state what their initiative is.
Are they supposed to roll initiative using their Dex modifiers (either individually or with their size class)? This would be as in the Find Familiar spell:
In combat, it rolls its own initiative and acts on its own turn.
Or do they act immediately after the caster? This would be as in the Summon Beast spell:
In combat, the creature shares your initiative count, but it takes its turn immediately after yours.
Or does this simply require a decision by the DM?
I have realized that one key difference between the spells I cited explains their different treatments of initiative: their casting times.
Find Familiar has a casting time of 1 hour, so it will never be cast during combat. Combat will have begun substantially after the spell was cast, which is why the familiar must roll initiative to determine how quickly it's able to react to the start of hostilities relative to everyone else.
Summon Beast and Animate Objects, however, both have casting times of 1 action, so their summoned creatures will typically appear mid-encounter. As such, those creatures should begin to act as soon as they arrive, as would be true for anyone who shows up to the battle mid-combat.