According to Web sources, the standard British and American helmet in World War I was invented by John Leopold Brodie, and was patented in 1915. He was a London-based engineer. It had been found that shell splinters rather than bullets were the main danger to soldiers' heads, and the design was meant to protect them from splinters (and hence from above).
However, the three British Brodie patents for helmets do not look anything like the classic flattish helmet with a broad brim. These vary somewhat, but tend to look like the one illustrated below.
The 1915 patent, GB 1915/11803, illustrated below, was applied for in August 1915. There was also US 1251959 for the same invention. See how much higher the dome is in the drawings. Also, the brim is at a greater angle of slope.
There was a lining made of "American cloth" which was integral to the helmet. An air gap between the lining and the helmet kept the head cool in summer while preventing frostbite in winter. It also prevented rusting; kept the helmet firmly on the head; and prevented pressure on any point on the head, which could cause headaches. Also in the patent, Brodie suggested that it be painted in rainbow colours so as to make it "invisible to the enemy", and early issues were indeed painted, apparently.
So why was it so different ? I can only speculate that a need to save metal meant that the flatter shape was adopted, assuming that the Brodie attribution is correct. It was first used in any numbers in July 1916.
There is a Wikipedia article on the Brodie helmet. To save time designing a new helmet, sources say, the US Army adopted a slightly modified form of the Brodie helmet for its soldiers.