How do we emphasize the urgency of a situation or immediacy of an event? How can we reassure someone that something will be taken care of right away?

The adverb gleich is used a lot and has many possible translations in this context. You might be thinking about it as an adjective meaning "same" or "alike," and we also have a lesson on that topic. But here we are looking at how it is used to define something as happening "in just a minute," "immediately," or "momentarily." 

Gut, vielen Dank. Ich mache mich gleich auf den Weg.

Good, many thanks. I'll be on my way immediately.

Caption 14, Berufsleben - Probleme mit Mitarbeitern

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Warum hab ich das nicht gleich erkannt?

Why didn't I realize that right away?

Caption 80, Die Pfefferkörner - Endspurt - Part 12

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Ja, Moment, ich habe es gleich.

Yes, one moment, I'll have it momentarily.

Caption 30, Der Merkelpilot - der kleine Mann, der es macht

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There is another way to say that something will happen "any minute now," and it's very similar to how it's said in English:

Aber der Reparaturdienst müsste jede Minute da sein.

But the repair service should be here any minute.

Caption 8, Die Pfefferkörner - Endspurt - Part 18

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Another adverb to know is sofort, which is even more urgent than gleich, and is pretty much always translated as "at once," "right away," or "immediately."

Kein Problem, Frau Schmidt. Ich werde es Ihnen sofort schicken.

No problem, Ms. Schmidt. I will send it to you immediately.

Caption 67, Berufsleben - das Vorstellungsgespräch

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Sie ist sofort abgehauen, als Frau Lenz die Sache angesprochen hat.

She immediately took off when Ms. Lenz raised the matter.

Caption 65, Die Pfefferkörner - Endspurt - Part 2

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In more formal contexts, you may hear unverzüglich ("without delay") and umgehend. 

Er muss unverzüglich Waffenstillstandsverhandlungen mit Eisenhower und Montgomery aufnehmen.

He must immediately begin armistice negotiations with Eisenhower and Montgomery.

Caption 81, Die Stunde der Offiziere - Dokudrama über den 20. Juli 1944

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Oder noch besser, komm umgehend ins Kontor.

Or better yet, come to the office immediately.

Caption 65, Die Pfefferkörner - Alles auf Anfang

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Although unmittelbar is also translated as "immediate," it is often related to space rather than time: 

Wir sind, äh, hab ich schon gesagt,

We are, uh — as I've already said —

in unmittelbarer Nähe, äh, von Saarbrücken.

in the immediate vicinity, uh, of Saarbrücken.

Captions 37-38, Nachbarn - Andrea aus Forbach

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Daher ordne ich an, dass Sie unmittelbar wieder die Ehe schließen.

Therefore I order you to remarry immediately.

Captions 34-35, Yabla Gerichtshof - Probleme des Zusammenlebens

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For urgency, there is the word dringend, which like many German words functions as both an adjective and an adverb depending on how it's integrated in the sentence: 
 

Ich brauche dich hier dringend im Innendienst.

I need you urgently here in the office.

Caption 63, Großstadtrevier - Von Monstern und Mördern

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Further Learning
You will find many examples of these words used on Yabla German. In particular, keep an eye on how the translation of gleich changes depending on the context.