Let’s say that you heard the name of the Prophet (blessings and peace of Allah be upon him) then you did not send blessings upon him straight away. What is the ruling on that?

Praise
be to Allah

Firstly: 

The scholars differed as to whether it is obligatory to send
blessings upon the Messenger of Allah (blessings and peace of Allah be upon
him) when his name is mentioned. 

Ibn al-Qayyim (may Allah have mercy on him) said:

There is a difference of opinion as to whether it is
obligatory to send blessings upon him (blessings and peace of Allah be upon
him) every time his name (blessings and peace of Allah be upon him) is
mentioned. Abu Ja‘far at-Tahhaawi and Abu ‘Abdullah al-Haleemi said: It is
obligatory to send blessings upon him (blessings and peace of Allah be upon
him) every time his name is mentioned. 

Others said that this is mustahabb but it is not obligatory
in the sense that the one who does not do it is sinning. 

Then they differed further. Some said that it is obligatory
to send blessings upon him (at least) once in a lifetime, because the
command in general terms does not imply that it is to be done repeatedly,
and it may be attained by doing it once. This is narrated from Abu Haneefah,
Maalik, ath-Thawri and al-Awzaa‘i. ‘Iyaad and Ibn ‘Abd al-Barr said: It is
the view of the majority of the ummah. 

Others said: rather it is obligatory to do that in every
prayer, in the final tashahhud. This is the view of ash-Shaafa‘i, of Ahmad
in the latter of the two reports narrated from him, and of others. 

Yet others said that the command to send blessings upon the
Prophet (blessings and peace of Allah be upon him) is a command in the sense
of it being recommended, not in the sense of it being obligatory. 

End quote from Jala’ al-Ifhaam (p. 382) 

The hadiths that speak of supplication against (one who does
not send blessings upon him and which describe as miserly and hardhearted
the one in whose presence the Prophet (blessings and peace of Allah be upon
him) is mentioned and he does not send blessings upon him, support the view
of those who say that it is obligatory to send blessings upon the Prophet
(blessings and peace of Allah be upon him) every time his name is mentioned,
in general terms. 

The Messenger of Allah (blessings and peace of Allah be upon
him) said: “May his nose be robbed in the dust, one in whose presence I am
mentioned and he does not send blessings upon me.” Narrated and classed as
hasan by at-Tirmidhi (3545); classed as saheeh by al-Albaani in Saheeh
Sunan at-Tirmidhi. 

It was narrated from al-Husayn ibn ‘Ali ibn Abi Taalib (may
Allah be pleased with him) that the Prophet (blessings and peace of Allah be
upon him) said: “The miser is the one in whose presence I am mentioned and
he does not send blessings upon me.” Narrated and classed as hasan by at-Tirmidhi
(3546); classed as saheeh by al-Albaani in Saheeh Sunan at-Tirmidhi. 

Al-Faakihaani (may Allah have mercy on him) said: 

The hadith “The miser is the one in whose presence I am
mentioned and he does not send blessings upon me” supports the view of those
who say that it is obligatory to send blessings upon him every time he is
mentioned, and this is the view towards which I am inclined.

End quote from al-Qawl al-Badee‘ fi’s-Salaah ‘ala al-Habeeb
ash-Shafee‘ (p. 31) 

This is also the view of a number of scholars, including: at-Tahhaawi
among the Hanafis; at-Tartooshi and Ibn al-‘Arabi among the Maalikis; Abu
‘Abdullah al-Haleemi and Abu Haamid al-Isfaraayeeni among the Shaafa‘is; and
Ibn Battah among the Hanbalis. 


Al-Mawsoo‘ah al-Fiqhiyyah
(1/204). 

See also the answer to question no.
128796 and 131667. 

Secondly: 

According to the view that it is obligatory to send blessings
upon him (blessings and peace of Allah be upon him) every time his name is
mentioned, it is required of the one who hears his name (blessings and peace
of Allah be upon him) to send blessings upon him immediately, without any
delay, because this is an act of worship that is connected to a specific
time, and becomes obligatory at that time, and is missed if the time for it
is missed. See the answer to question no. 145693. This is supported by the
apparent meaning of the hadith quoted above: “May his nose be robbed in the
dust, one in whose presence I am mentioned and he does not send blessings
upon me” 

As-Saalihi (may Allah have mercy on him) said: 

The sending of blessings upon him should come immediately
upon mention of his name in one’s presence; if a person delays doing so, he
is deserving of blame.

End quote from Subul al-Huda wa’r-Rashaad (12/421). 

If the interval between mention of the Prophet (blessings and
peace of Allah be upon him) and the sending of blessings upon him is
lengthy, then it is an act of worship for which the time has passed, and one
has missed out on doing it as a result. 

But if the interval is brief, then it does not matter. 

If the individual forgot and a long time has elapsed, then he
remembers and sends blessings upon the Prophet (blessings and peace of Allah
be upon him), then there is nothing wrong with that either.

This is like the adhkaar that are recited after the prayer;
it is Sunnah to recite them immediately after the prayer. If a long time has
elapsed, then the time for reciting them has been missed, but if only a
short time has passed, it does not matter. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: 

If a long time has elapsed between the prayer and the dhikr,
then the time for reciting it has been missed. What is regarded as a long
time is a matter of custom and convention [i.e., there is no specific
definition; rather reference is to be made to what is customarily regarded
as a long time]. But if only a short time has elapsed, which includes
performing the funeral prayer, then the time has not been missed.

End quote from Sharh ‘Umdat
al-Ahkaam 

For more information, please see the answer to question no.
148718

Conclusion: 

The one who loves the Prophet (blessings and peace of Allah
be upon him) should be keen to send blessings upon him (blessings and peace
of Allah be upon him) straightaway every time he hears his name, and he
should not delay doing that.

See also the answer to question no. 68837

And Allah knows best.