Praise be to Allah.
The Muslim has to understand the importance of prayer and the necessity of observing prayer regularly at the appointed times, fulfilling all the conditions and pillars and essential parts of the prayer. Allah says (interpretation of the meaning):
“Verily, As-Salah (the prayer) is enjoined on the believers at fixed hours.”
It was narrated that ‘Abd-Allah ibn Mas’ood (may Allah be pleased with him) said: I asked the Prophet (peace and blessings of Allah be upon him), “Which deed is most beloved to Allah?” He said, “Prayer performed on time.” He said, “Then what?” He said, “Honouring one’s parents.” He said, “Then what?” He said, “Jihad (fighting) for the sake of Allah.”
Narrated by al-Bukhari (504) and Muslim (85).
So it is not permissible for the Muslim to delay his prayer beyond the time it is due. Wudoo does not require a clean place. Even if we assume that it does, the questioner could make sure that he has wudoo before coming to work and keep his wudoo so that he can pray on time.
He has to offer prayer on time. Looking for a clean place where he can perform the prayer is not too difficult. Prayer is valid in any place on earth so long as it is clean and pure. The Prophet (peace and blessings of Allaah be upon him) said: “The earth has been made for me a place of prayer and purification, so whenever the time of prayer comes for any man of my ummah, let him pray.” Narrated by al-Bukhari, 335; Muslim, 521. Sharee’ah (Islamic law) only excludes from this specific places in which prayer is not valid, which include graveyards and bathrooms. The Prophet (peace and blessings of Allah be upon him) said: “All the earth is a place of prostration apart from graveyards and bathrooms.” Narrated by Abu Dawood, 492; classed as saheeh (authentic) by al-Albani in Saheeh Abi Dawood.
So the workplace may be a place of prayer if it is clean. If it is not easy then he should look for somewhere else. Asking his bosses to allocate a corner where he can pray is not something that is difficult.
Shaykh al-Islam Ibn Taymiyah said:
If he deliberately delays the prayer until the time is too short to meet all the conditions and do all the essential parts of the prayer, such as if there is some impure substance on him, or he is junub and delays the prayer so that if he does ghusl the time for prayer will be over, then he has to purify himself first, but he is a sinner because he is doing the prayer at the wrong time. He should have purified himself before the time became too short and prayed on time. When he delays it, then he has to do what is required in the right sequence even though he is a sinner because of his delaying it.
If he is unable to fulfil one of the conditions of prayer being valid – such as taharah (purity) – then he should pray on time and he is exempted from this condition. It is not permissible to delay the prayer until its time is over in order to fulfil that condition.
Shaykh al-Islam Ibn Taymiyah said:
Because those who are obliged to pray when the time for prayer comes, but are unable to fulfil one of its conditions or essential parts at that moment, but will be able to do so after the time for prayer is over, it is not permissible for them to delay the prayer until its time is over. If this were permissible then the one who is unable to purify himself, cover himself, bow or prostrate, or do any other condition or essential part of the prayer would be allowed to delay the prayer until he is able to do this, if he knows or thinks it most likely that he will be able to do it. This is contrary to the Quran, Sunnah and scholarly consensus, because sharee’ah’s concern that the prayer be performed on time is greater than its concern for any other condition or essential part of prayer that cannot be done. Hence it is not permissible to delay the prayer until its time is over because of being unable to do some of the essential parts. When the time for the obligatory prayer is too short and one cannot fulfil the conditions, then doing the prayer on time takes precedence over the conditions. There is only one time when fulfilling the conditions takes precedence over the act itself, which is when the act becomes obligatory towards the end of its time, such as when a sleeper wakes up at the end of the time, in which case the prayer becomes obligatory when he wakes up, and he must fulfil the conditions of prayer. This applies if someone wakes up after the time for prayer is over.
And Allah knows best.