Fiona Bruce — Sex-obsessed. For a clean quote: "I'm Fiona Bruce. Breaking news, breaking hearts." On the radio this role was taken by the popular Radio 4 newsreader and continuity announcer Charlotte Green.
The FourthDoctor — One of Culshaw's best known impressions. He's also done a few of the other Doctors as well.
He rang the various actors who had played The Doctor over the years. This lead to some very amusing quotes, including:
Culshaw:"Hello, this is The Doctor." Tom Baker:"That's odd... Oh no, no, there must be a mistake, I am the Doctor."
In fact, he's done almost every Doctor, as we saw when he appeared on one of the BBC's Doctor Who retrospectives. He's not done John Hurt and Peter Capaldi yet, but give him time.
David Cameron — as a vote-grabbing popularist who prefers to avoid controversy.
Sir Menzies Campbell (At the time, leader of the Liberal Democrats) — a doddering old man who liked to sleep a lot. Needless to say, this one was a bit on the controversial side. This impression was retired after he stepped down.
The BBC Radio 4 continuity announcers (back when the show was on the radio), particularly Brian Perkins, portrayed as a gangland boss fond of torturing and dispatching those who upset him. In the TV series, Michael Buerk took over this role.
Anachronism Stew: The radio version had a sketch supposedly celebrating the 700th anniversary of The Today Programme, with a look back at the first, which was set in the Hundred Years' War but nonetheless had expies of modern presenters and politicians talking in Ye Olde Butcherede Englishe. This proved popular enough to get a TV adaptation (which used Newsnight instead).
George W. Bush: My fellow... ("Invertebrates", "Umbrellastands", anything but "Americans".)
John Humphries: And the time is coming up to... (some bizarre number or concept like 'eleventy-umpteen past banana' - referring to how Humphries famously would read the time out incorrectly and convince millions of drivers they were late for work)
Cloudcuckoolander: The Doctor, at least in how he is perceived by all the people he rings up.
Cowboy Bebop at His Computer: Unlike the well-researched Star Trek/Star Wars/Doctor Who sketches, the TV takes on the films of The Lord of the Rings and Harry Potter tended to fall into this - for example, Frodo's said to be on a quest to find the One Ring, and the Harry Potter and Voldemort impressions sound nothing like those in the films. (The radio edition's voices for the Harry Potter characters predated the first film, and they stuck with those even after it was released.)
Dreadful Musician: The Sing Something Simple Singers were regularly featured on the show's version of Crimewatch for "murdering hundreds of innocent songs".
Dropped a Bridge on Him: When Matt Smith (not that one) left Go 4 It, the radio version interpreted this by having him told by his producer to "investigate what it's like inside a sack", after which he was bundled into a van and thrown into a river.
I'm drowning! Only joking. No, I'm not. (Glub, glub, glub.) ... ... ...Ace.
Fetch Quest: Gandalf keeps sending Frodo on epic quests... to fetch thinks like milk and cigarrattes ("But don't tell Bilbo! He thinks I've given up.")
Ham-to-Ham Combat: A recurring sketch has Ian Mackellan and Alan Rickman fighting over who gets to be the resident Evil Brit, ultimately resulting in them shooting each other to death. Literal ham-to-ham combat.
Hidden Depths: David Beckham was depicted as being fairly dim off the football pitch, but displays occasional moments of cutting insight or cultural depth.
Hilarious in Hindsight: The Fourth Doctor suspected there was something alien about the London Eye. When Doctor Who returned, the first episode involved the Nestene Consciousness using the Eye as a transmitter.
Prank Call: The radio version did a few of these: the Fourth Doctor phoning (real) Directory Enquiries in search of the Master is particularly memorable, especially the one where the operator suggested that he might need the international directory.
Brian Sewell: No. Eastenders is a cavalcade of parsimonious bleating, bereft of cultural significance on every conceivable level.
Roving Reporters: D'you like chips?
Brian Sewell: Bugger off!
Shown Their Work: The science fiction references were pretty much all well-researched, especially the Doctor Who ones. All of the Fourth Doctor's jargon refers to actual planets, technologies and so forth not only from the show, but from the period of his tenure.