People are over exaggerating this. Its really this simple.
British soldiers say leftenant, because they do. They always have. Most likely always will. It spans century after century of rich military history, achievements and battle honours. Like so many other British military traditions and preserved ways of doing things, we will probably never truly know why the word is used that way. There is most likely no language definition that can cover it. Quote all the Latin, French or Eskimo that you like, it makes no odds. Its probably just become that way over time as it rolls off the tongue easier. In effect, it is military slang. That is how the British army says it. That's the fact here.
Americans just say lieutenant. Because that's how its spelt.
For informative value, i have read that the 'left' part is closer too the original French spellings of the word used around the 17th century and it may have originated from that, along with the fact many French originating words were adapted by us Brits over the centuries because we don't like the French. Seems legit. Finally i also read that traditionally, lieutenants would stand behind and to the left of their commanding officer in battle. I am a former soldier, and i am unsure of this last one, i find it a bit unlikely, unless every commanding officer stood on the far right of his entire regiment in a battle......but again its just for information's sake.
What you should really be asking, is why Americans say aluminum and not aluminium.