I think I can answer what's happened with Oman there. While the Omanis did have some conflict with the Qawasim (Ras al-Khaimah) and the Wahhabis to the north, they didn't conquer that territory as the map shows. (Though they had controlled the island of Hormuz and Bahrain briefly in the 1800s.) Actually in 1920 the British negotiated the Treaty of Sib which formally divided control of Oman itself between the Sultan in Muscat and the Imam of Oman who ruled the interior, so Oman wasn't even really one country at this point. What the mapmaker has done is represented all the trucial states of the Gulf as 'Oman'. These sheikhdoms - modern day UAE - had their foreign relations managed by the British since 1892, with Qatar being added after the first world war. Sometimes the confusing term "trucial Oman" was used for this colllection of British protectorates, which possibly led to what's shown on this map. The Border between Saudi Arabia and Oman and UAE wasn't defined until later, so the mapmaker seems to have just drawn a line from Qatar to Dhofar and called it a day.

For the modern history of Oman and the Gulf:

• Commins, David Dean. The Gulf States : A Modern History. London: I.B. Tauris, 2012.

• Jones, Jeremy, and Ridout, Nicholas Peter. A History of Modern Oman. 2015.

As for Yemen, that is the location of the historical kingdom of Yemen and the Ottoman yemen vilayet before that. But the shapes of Yemen and the Aden protectorate are a bit wonky here, I think.

Edit: Another historical oddity you can spot on this map are some of the never-realized provisions of the 1920 Treaty of Sèvres, independent Kurdistan and the enlarged Armenia controlling Erzurum and the lake Van region. You can also see that Greece controls the Izmir/Smyrna region which would end in 1922.

The borders between Yemen and Saudi Arabia were established only (and not completely) in 1934 with the treaty of Taiz. We need to go back a bit in history and consider that during the Ottoman rule at the end of the XIX century, different areas of the Arabian peninsula were mostly controlled by different clans. From the XIX century, the clan of the Al Saud from the Emirate of Najd (an area in modern day Saudi Arabia) had started to unite vast territories in the peninsula and rival with the declining Ottomans. The Ottomans left after the First World War and what we see in the map here in 1920 as Yemen was the territory controlled by the Zaidi Imam, that includes the core of future North Yemen. Aden was a British protectorate and Hadramaut at the time wasn’t officially controlled by the British Empire but technically it was. Asir was a region that both Saudi Arabia and Yemen wanted, but in 1920 it was controlled by the Idrisi clan, a Sunni clan originally from modern day Libya allied with the British. Additionally, both Yemen and Saudi Arabia claimed the area of Najran. Saudi Arabia and Yemen clashed till 1934 for the control of the territories of Asir, Najran and Hodeida. We have to consider that Saudi Arabia was already becoming a regional power and it was more powerful than Yemen, and it managed to occupy the port of Hodeida before the treaty was signed. With the treaty of Taiz in 1934, Yemen got Hodeida but had to give to Saudi Arabia parts in the north and Asir was officially declared part of Saudi Arabia. Till today, Yemen has claims on territories in both Asir and Najran. Hejaz became part of the Al Saud territories in 1932 but to be honest I don’t know much about the story behind it apart that it was controlled by a different clan. We have to take into account, in any case, that in 1920 the borders between the Arabian peninsula countries weren’t well established. What we see in green in this map is the territory of the Imam Yahya, in what will become North Yemen, while South Yemen will include the British protectorate of Aden, Hadramaut and the island of Socotra, another British protectorate in this map. Modern Yemen became a single country only in 1990.

I hope I was clear enough, English isn’t my mother language.

Source: Storia dello Yemen, F. Sabahi