What's the correlation between 済む and すみません? 済む means to finish and I can't see it a correlation between the two.
A better understanding of 済む is to be settled or be cleared off. Thus すみません represents the sentiment that one owes a mental debt, something remain to be repaid. Here comes the meaning "I feel guilty to you", or "I'm sorry".
FYI the negative of 済む as such has that lexical meaning besides this cliche.
= I did something conscience-smitten to him
≈ "I regret what I did to him"
While there are some that would dismiss this as a false correlation, there is a connection. According to jisho.org すみません can also be written as 済みません, but there is an important note in the definition. It is usually written in kana. Personally, before the research I had never seen it with kanji before.
済む has a variety of meanings, one of which is
to finish as you have noted. However, the fourth definition,
to feel unease or guilt for troubling someone; to be sorry, is more connected in meaning to すみません (notice the subtext next to the definition in the link).
It also turns out that the conjugation of 済む into すまない (usually written in kana alone) is also related to すみません. (See definition 3)
When it comes to すみません, it's best not to overthink these connections. No matter the connection, this set phrase will always mean
excuse me; (I'm) sorry or
thank you depending on context.