Amazing how a whole country can be wrong.
Soldering is something you do with a soldering iron.
Soldering and brazing were in use 2500-3000 years ago. The Romans used lead/tin solder in plumbing.
The English *word* comes from Latin solidare or solidus which clearly have pronounced 'l's. The verb soldering is recorded by 1466 (Paston Letters ... Norfolk gentry), before the Americas were (re)discovered.
Just one small problem with your etymology: it is incorrect. English got the word not from Latin, but from French, by which point the L had already been dropped. Then,
after the US colonies were established, English in the UK underwent a wave of “re-latinization”, during which the L was added back. And that change never propagated to the US.
It's not unusual for pronunciation to drift in isolated communities such as the USA.
As a general statement, that is true. However, in the specific case of English, it is actually British English that drifted far more. This is well established by historical linguists of English (for example, the British linguist David Crystal, who has written several definitive works on the history of English, published by Cambridge University Press.)
The best summary of the situation is this: since British English and American English “split”, both dialects have evolved away from the common ancestor, but British English evolved away
more. So where the dialects differ, there is a greater chance that American English retains, or is closer to, the original form.
(I studied linguistics at university, so I have a deeper background in this than the average Joe.)