Why do the English say “leftenant” and the Americans say “lootenant” when the spelling of “lieutenant” indicates a pronunciation like “lyewtenant”?

The American “lootenant” is easy: it’s a general rule of American English that [ju] after alveolar consonants is reduced to [u]: news, tune = nooz, toon. In British English, they are nyooz, tyoon. (And there is variation within American English.)

The lack of a French pronunciation is also regular: French ieu is rendered in English as -(y)oo. Thus in lieu of; adieu.

The oddity is the left- pronunciation. This is what the Oxford English Dictionary says:

The origin of the β type of forms (which survives in the usual British pronunciation, though the spelling represents the α type) is difficult to explain. The hypothesis of a mere misinterpretation of the graphic form (u read as v), at first sight plausible, does not accord with the facts.

I.e. that people misread lieutenant as lievtenant.

In view of the rare Old French form luef for lieu (with which compare especially the 15th cent. Scots forms luf– , lufftenand above) it seems likely that the labial glide at the end of Old French lieu as the first element of a compound was sometimes apprehended by English-speakers as a v or f.

Meaning, the unfamiliar French liøtenã, with a breathy pronunciation, could have been misheard in England as lyeftenant, a spelling present alongside leuetenant since the Middle Ages.

Possibly some of the forms may be due to association with leave n.1 or lief adj.

i.e. Folk etymology.

In 1793 Walker gives the actual pronunciations as /lɛv-//lɪvˈtɛnənt/, but expresses the hope that ‘the regular sound, lewtenant’ will in time become current. In England this pronunciation /ljuːˈtɛnənt/ is almost unknown. A newspaper quot. of 1893 in Funk’s Standard Dict. Eng. Lang. says that /lɛfˈtɛnənt/ is in the U.S. ‘almost confined to the retired list of the navy’.

So the old, predictable pronunciation had died out in England; the spelling survived, and it may be that either the old pronunciation survived in the US, or was revived as a spelling pronunciation.