He was going for Cockney but, as always with Neeson, his accent is at least 25% Northern Irish.

I thought he was trying to pull off some sort of northern or perhaps mancunian accent specifically. He sounded a lot like Christopher Eccleston to me.

I thought it was supposed to be Cockney, but it was about as convincing as Keanu Reeves in Bram Stoker's Dracula.

I thought he was going for an Aussie accent. Or maybe he was a cockney who'd spent a long time in Australia. It's a hard accent to place but it's not completely implausible. Cockney-Australian hybrid accents seem to be particularly popular among music-video directors.

I'm just watching this movie for the first (and probably only) time, and when I saw the funeral scene with Liam, I got to thinking: when Liam talks in his accent in that scene, he sounds similar to Christopher Eccleston's accent in Gone in 60 Seconds.
Maybe it's just me

I agree with that, although I'm uncertain if that's what he was trying, and even less certain what his accent actually was.
Dave (The Werebeagle)
If that's my nose, why does it look like your thumb?

I'm guessing it's supposed to be an Australian accent, definitely not a London accent (I'm from there).

This was at a period when any kind of English accent indicated you were some kind of creative genius!
Remember the disaster of getting Mike Sarne to direct "Myra Breckenridge" ?
Anywhoo, Neeson was TRYING to assay some kind of London accent, not unlike Mick Jagger (note he repeatedly calls Callahan "love")
It's a step up from Dick van Dyke and "Gorblimey, Mary Poppins" !!
Come on lads, bags of swank!