nachträglich
belated
Taking place or happening at a later point in time, after the actual event has already passed.
Herzlichen Glückwunsch nachträglich zum Geburtstag!
Happy belated birthday!

belated / subsequent
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belated
Taking place or happening at a later point in time, after the actual event has already passed.
Herzlichen Glückwunsch nachträglich zum Geburtstag!
Happy belated birthday!

The word is made up of 'nach' (after) and a form of 'tragen' (to carry). It therefore describes something that is 'nachgetragen' (added later), meaning added or made up for later.
Die fehlenden Informationen wurden nachträglich per E-Mail geschickt.
The missing information was subsequently sent by email.
As an adverb, 'nachträglich' usually stands in the middle of the sentence, often after the conjugated verb and temporal information. It indicates that the action of the verb takes place later than the main event.
Wir haben die Dokumente erst gestern nachträglich erhalten.
We only received the documents subsequently yesterday.
In this meaning, 'nachträglich' is very often used for belated congratulations or apologies. Typical phrases are 'nachträglich alles Gute wünschen' (to wish someone all the best belatedly) or 'sich nachträglich entschuldigen' (to apologize afterward).
Ich möchte mich nachträglich für mein Verhalten entschuldigen.
I would like to apologize afterward for my behavior.
This meaning is often found in formal, technical, or legal contexts. Signal words are verbs like 'ändern' (to change), 'einbauen' (to install/build in), 'genehmigen' (to approve), or nouns like 'Vertrag' (contract), 'Protokoll' (minutes/record), and 'Gebäude' (building).
Der Balkon wurde nachträglich an das Haus angebaut.
The balcony was subsequently added to the house.
beforehand
'vorher' and 'nachträglich' are opposites in terms of time. 'vorher' means that something happens before a certain event. 'nachträglich' means that something only happens later, when the actual event is already over. It can also mean that something is added later, for example in a document. The words can be confused because both compare an action with another point in time. Pay attention to the direction: 'vorher' comes before, 'nachträglich' comes after.

Der Chef trägt die fehlenden Stunden nachträglich in die Liste ein.
The boss adds the missing hours to the list afterward.
Bezahl bitte vorher, damit wir an der Kasse schneller fertig sind.
Please pay beforehand so that we can finish more quickly at the checkout.