Speaking as someone who was once a PhD student in Linguistics: the question is really "why do people in the UK pronounce it "soLder?"
It's very common to have a "silent l" (so to speak) before oral stops (b, p, t, d, g, k) in English. Think about words like "would, could, walk, talk, yolk, etc." The "l" in all of these words is not normally pronounced. By analogy, no one should be surprised that "solder" is pronounced as "sodder"
The (mostly British) pronunciation of the "l" is basically a conscious "overcorrecting" This overcorrecting would (haha) be hard to apply to extremely common words, which is why "would, could" are never pronounced with an "l" sound.
But for semi-common words that you don't hear very often (and see mostly in written form), it's easy to start pronouncing them "phonetically' because you aren't entirely sure how they are pronounced. "Folk" and "yolk" (semi-uncommon words) are good examples - some people do pronounce the "l" .
I would bet serious money that non-technical speakers on both side of the Atlantic (or Pacific) would say "soLder" when they first see that word for the first time.
So again, the question is really "Why do British people artificially pronounce the l? Sometimes the answer is societal, not linguistic: if there were a famous song, a book, etc. in the UK where the person said "soLder" then that might have influenced other speakers to also pronounce it the same way.
And if you think that's a stretch - I can hear a clear inter-generational difference in the way Americans pronounce the word "frontier" - those who grew up with the original Star Trek and those who did not. Thank you again William Shatner for permanently scarring my pronunciation of this word
