I have no idea what "taking the L" means & I have a reasonably good knowledge of Australian slang. When I googled "taking the" the first five listed options were: ( taking the ) piss, next step, mickey, fifth & pill.....so taking the L doesn't even register as a common usage extension from "taking the"
When I put in "taking the L", the results were..(taking the) long way round, long way, lead, long way home & liberty.
If it's " taking the liberty " then the more commonly used form of that in "cockney" (English) slang is "taking a liberty", not " taking the liberty"
A slang phrase or expression might have limited, esoteric use amongst a particular group somewhere, eg a particularly juvenile part of the internet, but that's quite different from being "common usage"
Cockney slang has spread to a much wider range of users than just those who live in part of London through migration, film, TV and books and cartoons.