I admit I'm not too familiar with archaeology, but from what I understand, issues of original ownership and repatriation can get quite murky when it comes to items that were plundered in ancient times (or left somewhere by a ruling power such as Persia or Greece), long before the colonialism of the last few centuries and the formation of today's nation-states, not to mention all the shifts in boundaries, the population movements, changes in cultural identity, and so on, that take place within any given region over the course of thousands of years.
In-universe, the Ark had been in Egypt for 3,000 years until it was uncovered in 1936. 1936 was also the year of the Anglo-Egyptian Treaty as the Kingdom of Egypt sought to reduce British influence. Some cursory reading indicates that at that time, any antiquities excavated within Egypt's borders were supposed to be considered property of the Egyptian government. If formal legal frameworks had actually been adhered to in the Tanis Dig depicted in "Raiders," then the excavators would have been reporting any and all finds to the Egyptian government. And if they had known about the Ark, Egypt likely would have claimed it as a piece of their own ancient history. Though I surmise that there still may have been some British bureaucracy involved during the transitions resulting from the Treaty (British troops wouldn't completely leave Egypt until 1956).
Kor