I used to start on Sunday. This meant the topics could be about as up-to-date as possible, but the downside was that I'd be rushed to get two decent sketches done, and this left me less time for rewriting.
Now, I start Friday by scouring for stories, and add any that have potential to a text document. This takes about 15 minutes.
Saturday, I'll do another pass on the stories (another 10 minutes), and do a mind-map on any that seem to have sketch potential. By the end of Saturday, I'll have two sketches planned, with the premise and second-twist decided, but no jokes or actual dialog yet written. Maybe an hour of work, all-in-all.
Sunday is where I do the main body of writing. Because I've been planning for a few days, I find the process straightforward. It's like my primed brain has already been working on jokes behind the scenes, just waiting for me to sit at my keyboard. First drafts will also be rough, but being prepared early gives me lots of time to come back and re-write. Remember: good writing is re-writing. Sunday's writing takes me a couple of hours, I guess, but it's spread out through the whole day.
First thing Monday morning, I will give the sketches a read through and tweak; and then again at about 11:00-ish, when I finally submit.
I won't even think about one-liners until Monday evening, but I already have a lot of stories sourced to give me the raw material. And I keep an eye on the news in case anything crops up that I reckon is joke-worthy.
Don't worry so much about being cutting edge. Newsjack does cover the whole previous week, and in the past have even went back two weeks. My feeling is that news happening on the Friday onwards is fine.