He delayed the prayer until the time for it was over with no excuse; does he have to do ghusl?

Firstly: 

Prayer is the foundation of faith, the mark of the successful, the joy of the righteous. Allah has prescribed it for the believers, as He says (interpretation of the meaning): 

“Successful indeed are the believers.

2. Those who offer their Salaah (prayers) with all solemnity and full submissiveness.

3. And those who turn away from Al‑Laghw (dirty, false, evil vain talk, falsehood, and all that Allah has forbidden).

4. And those who pay the Zakaah.

5. And those who guard their chastity (i.e. private parts, from illegal sexual acts)

6. Except from their wives or (the slaves) that their right hands possess, – for then, they are free from blame;

7. But whoever seeks beyond that, then those are the transgressors;

8. Those who are faithfully true to their Amanaat (all the duties which Allah has ordained, honesty, moral responsibility and trusts) and to their covenants;

Prayer is the first thing for which a person will be brought to account on the Day of Judgement. The Prophet (peace and blessings of Allah be upon him) said: “The first thing for which a person will be brought to account on the Day of Resurrection is his prayer. If it is sound then he will have succeeded, but if it is not sound he will be doomed.” Narrated by al-Tirmidhi, 413, Abu Dawood, 864; classed as saheeh by al-Albaani in Saheeh Abu Dawood. 

Allah erases sins by means of prayer and raises people in status thereby. It is the last religious commitment to be forsaken and if it is neglected then all religious commitment is lost. See question no. 33694. 

My brother, adhere to regular prayer before you die… pray before the funeral prayer is offered for you. 

If you are already praying regularly, then continue to do so. If you have been negligent about prayer, then repent and seek forgiveness before it is too late. Whoever turns to Allah, Allah will turn to him. 

Allah says, according to the hadeeth qudsi: “If he draws near to Me a hand span, I draw near to him an arm’s length, and if he draws near to Me an arm’s length, I draw near to him a fathom’s length, and if he comes to Me walking, I go to him at speed.” Narrated by al-Bukhaari, 7405; Muslim, 2675. 

So repent sincerely to Allah, and Allah will accept your repentance. 

Secondly: 

Being careless about prayer and delaying it until the time for it is over with no excuse is a major sin. See question no. 47123. 

Some of the scholars are of the view that the one who omits a single prayer with no excuse until the time for it is over is to be viewed as a kaafir. See question no. 39818. This is the view on which the answer to your question is based, which is the command to the one who omits a single prayer with no excuse until the time for it is over to do ghusl.  

That is because it is prescribed for a kaafir who becomes Muslim to do ghusl, as it is also prescribed for an apostate who returns to Islam. 

Shaykh Ibn ‘Uthaymeen said in al-Sharh al-Mumti’ (1/202): 

If a kaafir becomes Muslim, he has to do ghusl whether he is new to Islam or had apostatized. 

“New to Islam” means one who formerly belonged to another religion all his life, such as a Jew or Christian or Buddhist, etc. 

“Had apostatized” means that he had been a Muslim, then he apostatized from Islam – we ask Allah to keep us safe and sound from that – such as one who did not pray, or thought that Allah has a rival or partner, or prayed to the Prophet (peace and blessings of Allah be upon him) seeking his help at a time of hardship, or prayed to someone else seeking his help in a matter in which he could not help. 

The evidence that ghusl is required in such cases is as follows: 

1 – The hadeeth of Qays ibn ‘Aasim, who said that when he become Muslim the Prophet (peace and blessings of Allah be upon him) ordered him to do ghusl using water and lotus leaves. Narrated by al-Tirmidhi, 605; classed as saheeh by al-Albaani in Saheeh al-Tirmidhi. So the basic principle with regard to this matter is that it is obligatory. 

2 – He has cleansed himself inwardly from the filth of shirk, so wisdom dictates that he should cleanse himself outwardly by doing ghusl. 

Some of the scholars said that he does not have to do ghusl, and they quoted as evidence the fact that no general command was narrated from the Prophet (peace and blessings of Allah be upon him), such as saying whoever becomes Muslim let him do ghusl, as he said, “Whoever comes to Jumu’ah, let him do ghusl.” How many of the Sahaabah become Muslim but there is no report that he told them to do ghusl or said, whoever becomes Muslim let him do ghusl. If it were obligatory it would have been well known because the people need to know about it. 

We could say that the first view, that ghusl is required, is stronger, because the Prophet (peace and blessings of Allah be upon him) issued a rule to one member of the ummah for which there is no reason to assume that it applied only to him, so it may be assumed that it applies to the ummah as a whole. The fact that the Prophet (peace and blessings of Allah be upon him) issued the command to one person does not imply that he did not issue the same command to others as well. 

With regard to the fact that it is not narrated from each one of the Sahaabah that he did ghusl when he become Muslim, the fact that there are no reports does not mean that it did not happen, because the basic principle is to do that which the Prophet (peace and blessings of Allah be upon him) commanded, and it is not essential for there to be a report from each person who did it. 

Ibn Qudaamah said in al-Mughni: 

If a kaafir becomes Muslim he has to do ghusl, whether he is new to Islam or had apostatized… this is the view of Maalik, Abu Thawr and Ibn al-Mundhir. 

It says in al-Mawso’oah al-Fiqhiyyah (31/206): 

The Maalikis and Hanbalis are of the view that a kaafir’s becoming Muslim necessitates ghusl. If a kaafir becomes Muslim, he must do ghusl. They did not make any distinction between a kaafir who is new to Islam and one who had apostatized. Ghusl is also required of an apostate who comes back to Islam. 

And Allah knows best.