I think it was on Friday Night Live with Ben Elton, when Harry Enfield was in full swing with Loadsamoney and Stavros. However, on this one occasion (I am sure he never repeated it) he squeezed in a sketch late in the show based on a paper boy and he wore a flat cap with a suitable accent (could have been barrow boy). The show was clearly running a little over time as Ben Elton hurriedly took over. I remember how very clever Harry put the sketch together, but alas the character never appeared again - he was probably just trying it out. Anyone remember... this is a tough one.
Harry Enfield sketch
I believe you're referring to "the black and white sketch" from Friday Night Live Episode 9, "a man out of time, from when Britain used to have a film industry".
Whilst it doesn't feature the rushed ending you describe, Enfield does play a Cockney pavement newspaper seller.
Many thanks Aaron, much appreciated. Now it's trying to find if it exists on YouTube.