It's almost certainly the flags of the United Nations, but not the post-war version that everyone is familiar with.

At the Arcadia Conference at the end of 1941 - that's the post-Pearl Harbor one where Churchill and the Imperial Staff came to Washington for a couple of weeks to work out the initial implementation of fighting the war with their American counterparts - FDR wanted some sort of formal term for the alliance of nations fighting the War. (And since it'll likely be asked, unlike the derivation of the term 'Second World War', I don't know of a source that has looked into detail about precisely when and how 'Allies' became the defacto informal term for that alliance.)

It's not quite clear how FDR coined the term 'United Nations' as the name of this alliance. He may have had some help (Welles or Hopkins are the most likely suspects for brainstorming; Sherwood would have claimed credit post-war, which he didn't), but to a degree it was a process of elimination. FDR was extremely vigilant about even tangentially negative links between the First World War and this one, and as such any terminology that had 'Associated' in it was out since the United States had come into the Great War as an 'Associated Power.'

However the moniker was created, it was cemented when he revealed it to Churchill, who loved it. It was a phrase used in a lengthy poem by Byron that the latter knew by heart ("Here, where the sword United Nations drew/Our countrymen were warring on that day/And this is much - and all - which will not pass away") which he recited from memory at the White House New Year's Eve dinner the night before the formal alliance was signed.

So on January 1, 1942, the initial 26 countries of the United Nations signed a statement declaring their cooperation against the nations of the Tripartite Pact. Here's a link to one of the 1942 posters showing the various Allies.

But why would there be 35 flags? Presuming the count is right (and I ain't going frame by frame to double check it), there were a number of other additions to the United Nations in 1943, 1944, and 1945. Here's a later war poster, and you can clearly see a number of countries included like Mexico and Brazil that were not original signatories. Those flags - although apparently not the last batch of signatories in 1945, who to a degree did so to gain access to the UN Conference in San Francisco - are clearly displayed on stage during the awarding of the letter.

If the date of the ceremony is correct, the one thing I can say with absolute certainty about the coalition represented is that it was not the post-war United Nations. It took place a about a week before a majority of states that had signed it back in June formally ratified the UN Charter, which was implemented on October 24, 1945.