Hvis leveringsdatoen ændrer sig igen, bliver jeg hjemme i eftermiddag.

Because hvis means if and introduces a condition that may or may not happen.

• Hvis leveringsdatoen ændrer sig igen = If the delivery date changes again
• Når means when, and it is used when something is expected or assumed to happen.

So here, the speaker is not saying the date definitely will change again—only that it might. That is why hvis is the natural choice.

Danish very often combines nouns into one compound word.

• levering = delivery
• dato = date
• leveringsdato = delivery date

Then Danish usually adds the definite article to the end of the noun:

• leveringsdato = delivery date
• leveringsdatoen = the delivery date

So Danish does not need a separate word like English the here. The ending -en already does that job.

Because Danish often uses a reflexive pronoun with verbs that describe something changing by itself.

• at ændre noget = to change something
• at ændre sig = to change

So:

• Jeg ændrer planen = I change the plan
• Planen ændrer sig = The plan changes

In your sentence, leveringsdatoen is not changing something else; it is itself undergoing change. That is why sig is used.

Here sig is a reflexive pronoun, roughly corresponding to itself, himself, herself, etc., depending on the subject.

In this sentence, it refers back to leveringsdatoen. But in natural English, we usually do not translate it literally:

• leveringsdatoen ændrer sig = the delivery date changes

So even though sig is there in Danish, you normally would not say the delivery date changes itself in English.

Because Danish often uses the present tense for future meaning when the context already makes the time clear.

Here the future meaning comes from:

• the condition hvis
• the time phrase i eftermiddag

So ændrer and bliver are present forms, but the sentence is understood as future.

This is very normal in Danish. English does something similar in the if-clause:

• If the date changes again...

Danish just extends that natural present-tense usage to the main clause as well.

Because Danish main clauses follow the V2 rule: the finite verb normally comes in the second position.

In this sentence, the whole hvis-clause comes first:

• Hvis leveringsdatoen ændrer sig igen = first element

So the finite verb of the main clause must come next:

• bliver = second position
• jeg = comes after the verb

That is why you get:

• Hvis ..., bliver jeg hjemme ...

If the sentence stood alone, without the initial clause, it would be:

• Jeg bliver hjemme i eftermiddag.

Not here. Blive has several meanings in Danish, and context decides which one is meant.

Common meanings include:

• become
  
  • Jeg bliver træt = I become tired
• remain / stay
  
  • Jeg bliver hjemme = I stay home

So blive hjemme is a very common expression meaning to stay home.

Because Danish distinguishes between location and direction.

• hjemme = at home
• hjem = homeward / to home

So:

• Jeg er hjemme = I am at home
• Jeg bliver hjemme = I stay at home
• Jeg går hjem = I go home

In your sentence, the speaker is staying in a place, not moving toward it, so hjemme is correct.

Igen means again.

In ændrer sig igen, it means the change is happening one more time.

Its position is natural because with a reflexive verb, the reflexive pronoun usually stays close to the verb:

• ændrer sig igen

That is the most neutral order here. You may also hear other placements for emphasis, but ændrer sig igen is the standard, most natural version.

In Danish, i eftermiddag is the normal way to say this afternoon.

Danish uses i with several time expressions of this kind:

• i morges = this morning / earlier this morning
• i formiddag = this morning / late morning
• i eftermiddag = this afternoon
• i aften = this evening
• i nat = tonight

So i eftermiddag does not mean in the afternoon in a general sense here; it refers to this particular afternoon.

Yes. You can say:

• Hvis leveringsdatoen ændrer sig igen, så bliver jeg hjemme i eftermiddag.

That is also correct and very common, especially in speech. The så helps mark the result part of the condition.

Without så, the sentence is still completely natural:

• Hvis leveringsdatoen ændrer sig igen, bliver jeg hjemme i eftermiddag.

The version without så can sound a little leaner or more written, but both are good Danish.

Yes, this comma is standard because the sentence begins with a subordinate clause, followed by the main clause:

• Hvis leveringsdatoen ændrer sig igen, bliver jeg hjemme i eftermiddag.

The comma marks the boundary between:

• the condition: Hvis leveringsdatoen ændrer sig igen
• the main statement: bliver jeg hjemme i eftermiddag

So the comma here is the normal and expected punctuation.