GG5 et al. have さっきゅう leading.

https://www.nhk.or.jp/bunken/summary/research/report/2011_10/20111002.pdf

そうきゅう is the more common reading among pretty much everyone except newscasters

@@ -10 +10 @@
-<reb>さっきゅう</reb>
+<reb>そうきゅう</reb>
@@ -15 +15 @@
-<reb>そうきゅう</reb>
+<reb>さっきゅう</reb>

i'm not 100% sold that these need to be two different senses.  none of my sources split

@@ -18,0 +19 @@
+<pos>&adj-no;</pos>
@@ -20 +21,4 @@
-<pos>&adj-no;</pos>
+<gloss>immediate</gloss>
+<gloss>prompt</gloss>
+<gloss>quick</gloss>
+<gloss>rapid</gloss>
@@ -24,9 +27,0 @@
-<sense>
-<pos>&adj-na;</pos>
-<pos>&n;</pos>
-<pos>&adj-no;</pos>
-<gloss>prompt</gloss>
-<gloss>quick</gloss>
-<gloss>rapid</gloss>
-<gloss>immediate</gloss>
-</sense>

Tanaka example:
１月２０日（月曜日）の午前中に最終の議事事項を出しますので、早急にお返事を下さい。 
We'll release the final agenda on the morning of Monday, January 20, so please reply quickly.
.
A customer wrote to me
早急に翻訳いただきありがとうございました。
She wasn't saying I did the translation urgently. (There was no indication of urgency.) 
She was saying I did it promptly, getting it back within minutes. (It was a short job.)
.
gg5
事件への早急な対応 a rapid response to an incident
・早急に immediately; urgently; promptly; at once; without delay
・対策を早急に検討する immediately consider countermeasures.
.
新和英中辞典
さっきゅう【早急】
¶早急に immediately; without delay [《文》 loss of time]; at once; straightaway; as quickly as possible
･早急に解決を要する問題 a matter that needs [demands] an immediate [a prompt] solution.
.
ルミナス和英辞典
そうきゅう1　早急
━[副] (急いで) in a hurry; (すぐに) without delay
━[形] (早急の) urgent, immediate

@@ -21,0 +22,10 @@
+<gloss>pressing</gloss>
+</sense>
+<sense>
+<pos>&adj-na;</pos>
+<pos>&n;</pos>
+<pos>&adj-no;</pos>
+<gloss>prompt</gloss>
+<gloss>quick</gloss>
+<gloss>rapid</gloss>
+<gloss>immediate</gloss>