What is required of one who completed his 'Umrah, but forgot to perform ablution before ṭawāf after his ablution was invalidated following the state of iḥrām?

Scholars have differed regarding the condition of purity (tahara) for Tawaf. The majority of scholars stipulate it, while Sheikh al-Islam Ibn Taymiyyah holds that it is not a condition, and that the Tawaf of one who is in a state of minor ritual impurity (hadath asghar) is valid. The majority relied on the hadith: "Circumambulating the House is a prayer." However, Ibn Taymiyyah weakened this hadith and considered it to be mawquf (attributed to a Companion and not the Prophet) from Ibn Abbas, and that Ibn Abbas's intent was devotion and remembrance during Tawaf.

Based on the view of the majority, if you performed Tawaf while in a state of ritual impurity, you remain in your state of Ihram and have not exited it. You must go to Makkah to complete the Tawaf, Sa'i, and shave or shorten your hair. You must immediately return to wearing your Ihram garments and refrain from its prohibitions. If you are unable to go to Makkah, your ruling is that of one who is prevented (muhsar). You exit Ihram by slaughtering a sheep in your place and shaving or shortening your hair with the intention of exiting Ihram. As for the prohibitions of Ihram that you committed out of ignorance, there is usually no sin upon you, and many scholars do not require a fidyah (expiation). However, it is more cautious to offer a fidyah for those prohibitions that involved damage (such as cutting hair). The fidyah is a choice between fasting for three days, or slaughtering a sheep, or feeding six poor persons.

Based on the second view (not stipulating purity), your Umrah is valid, but a hady (sacrificial animal - a sheep to be slaughtered in the Haram and distributed to its poor) is required. If you are unable to do so, then you must fast for ten days.