prompt
Happening immediately, right away, and without delay. Often used as an adverb.
Die Lieferung erfolgte prompt am nächsten Tag.
The delivery arrived promptly the next day.

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Happening immediately, right away, and without delay. Often used as an adverb.
Die Lieferung erfolgte prompt am nächsten Tag.
The delivery arrived promptly the next day.

'Prompt' (prompt/promptly) is almost always used as an adverb to describe that an action happens immediately and without delay. It answers the question 'How?'. Use as an adjective before a noun, e.g. 'eine prompte Antwort' (a prompt answer), is rare.
Auf meine E-Mail hat er prompt geantwortet.
He replied promptly to my email.
Although it resembles the English word 'prompt', the German word 'prompt' is only an adjective or adverb. It cannot be used as a verb, as in English 'to prompt someone' ('jemanden zu etwas veranlassen').
Ihre Frage veranlasste ihn zu einer schnellen Antwort.
Her question prompted him to give a quick answer.
Common verbs combined with 'prompt' (promptly) are 'antworten' (to answer), 'reagieren' (to react), 'liefern' (to deliver), and 'bezahlen' (to pay). It describes a quick and expected response.
Der Kundenservice reagierte prompt auf meine Beschwerde.
Customer service responded promptly to my complaint.
The word 'prompt' (prompt/promptly) is often used in more formal or business contexts. It sounds somewhat more refined than the everyday word 'sofort' (immediately).
Wir erwarten eine prompte Erledigung des Auftrags.
We expect prompt completion of the order.
fast / quick
'schnell' and 'prompt' can both mean that something happens without a long wait. 'schnell' is more general: it can refer to high speed, a short duration, or doing something quickly. 'prompt' more specifically means 'immediately' or 'without delay' and is especially suitable for reactions, replies, or tasks being dealt with. The words can be confused because both express a short amount of time. If you only mean speed, you usually use 'schnell'. If you want to emphasize that someone reacts immediately, 'prompt' fits better.

Auf meine E-Mail bekam ich prompt eine Antwort vom Kundenservice.
I promptly received a reply from customer service to my email.
In der Pause erklärt die Lehrerin die Aufgabe schnell noch einmal.
During the break, the teacher quickly explains the task one more time.