Dear respected Shaykh, assalamu alaikum wa rahmatullahi wa barakatuh. Is saying of a supplicating person: “sooner rather than later” considered as growing impatient which was prohibited by the Messenger of Allah (peace be on him) in his saying, “The invocation of anyone of you is granted (by Allah) if he does not show impatience? الاستعجال في الدعاء  

Praise be to Allah, and may Allah’s peace and blessings be upon the Messenger of Allah, his family and his companions.
Wa alaikum assalam wa rahmatullahi wa barakatuh.
To proceed:
In response to your question, we say, and with Allah Almighty lies all success:
What is apparent is that this does not fall under what was prohibited regarding  growing impatient while supplicating which is mentioned in a hadith transmitted by Bukhari (6340) and Muslim (2735) on the authority of al-Zuhri from Abu ‘Ubayd from Abu Hurairah (may Allah be pleased with him) who said that the Messenger of Allah (peace be on him) said: “The invocation of anyone of you is granted (by Allah) if he does not show impatience (by saying, “I invoked Allah but my request has not been granted.”
The Messenger of Allah (peace be on him) clarified what is meant by growing impatient or being hasty which impedes Allah’s response to a supplication. It is the saying of a person who is supplicating: “I invoked Allah but my request has not been granted.” It is narrated in Muslim (2735) on the authority of Abu Idrees al-Khawlani from Abu Hurairah (may Allah be pleased with him) who said the Messenger of Allah peace be on him said: “The supplication of a slave continues to be granted as long as he does not supplicate for a sinful thing or for something that would cut off the ties of kinship and he does not grow impatient.” It was said: “O Messenger of Allah! What does growing impatient mean?” He (the Messenger) said, “It is one's saying: “I supplicated again and again but I do not think that my prayer will be answered.” Then he becomes frustrated (in such circumstances) and gives up supplication altogether.”
In this hadith, The Messenger of Allah (peace be on him) clarified the meaning of hastening which is dispraised, which is a saying that includes feeling as if the supplication is a favour upon Allah, seeing response as being delayed and giving up supplication. Therefore the saying of a supplicating person, “sooner rather than later” does not fall under the prohibition. Abu Dawud (1159) narrates a hadith on the authority of Yazid al-Faqeer from Jabir Ibn Abdullah (may Allah be pleased with them) who said that when the Messenger of Allah (peace be on him) was making a supplication for asking for rain he said: “O Allah! Give us rain which will replenish us, abundant, fertilising and profitable, not injurious, sooner rather than later. He (the narrator) said: Thereupon the sky became overcast.”
And Allah knows best.

Your brother,
Dr. Khalid al-Mosleh
20 / 9 / 1427 AH