Was Wöhler’s urea synthesis carried out with or without oxygen?
Wöhler’s synthesis of urea is not from 1828 but from 1825. We’ll get to that in a second.

In cases like these, one should always rely on the original publications most. Unfortunately, publications of that era did not follow the somewhat stricter standards of experimental sections that nowadays’ publications do. Still, one can make very educated guesses based on the presence (or absence) of statements. The relevant quotation from the 1828 publication, in which Wöhler notes that the substance he created is indeed urea reads:[1]


  Ich habe schon früher angegeben, daſs man die oben erwähnte krystallisirte weiſse Substanz am besten erhält, wenn man cyansaures Silberoxyd durch Salmiak-Auflösung, oder cyansaures Bleioxyd durch flüssiges Ammoniak zersetzt.


Roughly translated: 


  I have previously disclosed that the best method of obtaining the above-mentioned white crystalline substance is by decomposing cyanacidic silver oxide[2] with salmiak or cyanacidic lead oxide with liquid ammonia.


It took a while to locate the previous reference which — in accordance with common custom at the time — was given as ‘In einer früheren kleinen Notiz, die in dem III. Bande dieser Annalen abgedruckt ist, …’ (‘In a previous little note that was printed in the IIIrd volume of these annals …’). In the end, I located the most probable candidate to be the artice Ueber Cyan-Verbindungen (On cyan compounds) also published in the Annalen der Physik.[3] Therein, Wöhler reports two syntheses of a novel crystalline compound; the second one being the interesting one for us:


  Rein erhält man [die eigenthümliche kryſtalliſierte Materie], wenn cyanſaures Blei durch kauſtiſches Ammoniak, oder cyanſaures Silber durch Salmiak zerſetzt wird. Sie kryſtalliſirt in weiſsen durchſichtigen, ſtrahligen Kryſtallen, iſt leicht in Waſſer und Alkohol auflöslich.


Translated roughly:


  [The weird krystalline material] can be acquired in pure form if cyanacid lead is decomposed with caustic ammonia or cyanacidic silver with salmiak. It crystallises in white transparent spear-like crystals and is readily dissolved by water and ammonia.


Wöhler then goes on to describe how different substances interact (or rather: don’t) with what he newly synthesised as was common in the days.

Here, we cannot really make any educated guesses on the reaction procedure by what is written because the fragments are extremely short. However, we can make educated guesses by what is not written. Wöhler made no reference to using sealed glassware or any specific atmosphere. He does not not any specific purification of the reactants. And he does not indicate any special care that may be required. Therefore, it is reasonable to assume that all reactions were performed at ambient pressure in the presence of oxygen and water (although water content will have been neglectable in the case of liquid ammonia).



References and Note:
[1]: F. Wöhler, Ann. Phys. 1828, 88, 253–256. DOI: 10.1002/andp.18280880206.

[2]: My note: Wöhler (incorrectly) assumed the acid anion to be $\ce{(CN)2O}$ which meant that the other constituent must have been silver(I) oxide giving $\ce{[Ag2O][(CN)2O]}$ — which gives the correct sum of $\ce{Ag(OCN)}$.

[3]: F. Wöhler, Ann. Phys. 1825, 79, 177–182. DOI: 10.1002/andp.18250790208.