Frankly, it sounds very "plebeian" in modern Polish, all those folksongs from old Czerniakow or Targowek suburbs.
(Performed by Stanislaw Grzesiuk for example.)
To start with, in linguistics, the term "accent" is technically incorrect if you ask about the difference in the sound "ch" in Ruch and Lech. That has nothing to do with the accent which in both words will be identical since there is only one syllable. Accent, in the context of the initial post, is colloquialism, a plebeian dialect of sorts in itself.
As such the term dialect would be more precise, and there is certainly such a thing as Warsaw dialect. Dialects often tend to sound "plebeian". Take Southern US, NYC, Cockney etc. They are certainly not mainstream and they sound plebeian.
In the song you linked to the accent is standard Polish. There are some faint traces of softening of some consonants, and I heard "£" that was on a dark side (although that alone is not a sure sign of the Warsaw dialect). But, the words being lyrics of a song I would hesitate to draw conclusions about the Warsaw dialect based solely on it.