The article notes that Alice's wedding gifts included "a magnificent diadem of brilliants" from her uncle and aunt, Tsar Nicholas II and Tsarina Alexandra Feodorovna of Russia. A separate article published a few days earlier in the New York Times notes that the gift given by the Tsar "is valued at a million rubles" (or $500,000 in 1903 US currency). The tiara wasn't the only gift given by Nicholas and Alexandra; he also gave the couple a motorcar, but it's not clear if the "million rubles" figure includes the tiara, the car, or both.
In my post, I speculate that the tiara given to Alice by Nicholas and Alexandra is the diamond tiara pictured above, which was later broken up to make Queen Elizabeth II's engagement ring and wedding bracelet. As Alice was only photographed (to my knowledge) in four tiaras...
1. her mother's star tiara, later lost in Russia 2. her wedding necklace set on a frame 3. the tiara pictured above 4. the meander tiara that now belongs to Princess Anne
... the tiara set with round diamonds (and, as some have speculated, perhaps aquamarines) seems to be the most likely possibility for Nicholas and Alexandra's gift.
My research on the question hasn't turned up any reliable sources of additional information, and I'd love to hear if anyone has further insights on the topic.