For political satirist Andy Zaltzman, it is the best of times, it is the worst of times.

Between the cloud cast by Brexit negotiations and the Twitter diplomacy of President Donald Trump, “there are a lot of balls in the air currently, on both sides of the Atlantic,” says the 43-year-old British comic. “Which makes it challenging and interesting from just a comedic perspective, but not always particularly relaxing to try and write about.”

Zaltzman is in the United States on a stand-up tour, with a Halloween stop in New York at the Gramercy Theatre. No matter the venue, Zaltzman says his audience tends mostly to be fans of his weekly podcast, “The Bugle.”

The “audio newspaper for a visual world” began in 2007 as a vehicle for Zaltzman and longtime collaborator John Oliver, now host of HBO’s “Last Week Tonight,” to keep working together after Oliver moved to the United States in 2006 to work on “The Daily Show.”

Oliver left “The Bugle” last year to focus full-time on his HBO gig.

With Oliver’s blessing, Zaltzman continued the smart, linguistically intricate satirical take on current events. The new “Bugle” has a co-host drawn from a rotating cast of diverse comics including Aussie Alice Fraser, Indian Anuvab Pal, American Hari Kondabolu and Zaltzman’s sister and fellow podcaster Helen Zaltzman.

“It was different when it was John and I,” Andy Zaltzman says, “because we had a comedic relationship that went back years before we started doing ‘The Bugle,’ so you can’t really replicate that. And I wouldn’t really want to.”

While the podcast offers “40 minutes to fill every week with whatever I want to talk about,” stand-up has something more. “It has a great instantaneous thrill to it, an excitement that you don’t get without a live audience.”

Although some “Bugle” features “eventually filter into stand-up,” one shared trait is puns. Minutes-long, love-it-or-hate-it “pun runs” cover anything from dog breeds to Latin writers to Russian cities to Charles Dickens novels.

Who is to thank, or blame, for these puns that used to drive Oliver into fits of (faux) outrage? Zaltzman says it’s his father’s fault. “My dad when I was a child always used to make lots of puns, so I guess that filtered through from him,” Zaltzman says, laughing. “It’s his poisonous legacy.”