Hello Cheeky was this to I'm Sorry I'll Read That Again. Both were sketch shows with a surreal bent, a cast that played exaggerated versions of themselves, No Fourth Wall, a shared actor in Tim Brooke-Taylor and, early on, a shared producer in David Hatch. The main difference was that Hello Cheeky was a lot quicker and looser, with sketches not needing a punchline. (Actually, the first season of Hello Cheeky ran parallel to the last season of I'm Sorry, I'll Read That Again, which resulted in playful potshots by the latter to the former.)
Not a programme, but an entire station: 96.3 Radio Aire in Leeds, West Yorkshire, could be considered as a spiritual successor to 103.2 Power FM in Hampshire. Similar playlist, emphasis on personality/music, rather than "more music less talk", and emphasis on dance music at the weekends, whilst retaining an air of locality; in fact they only network 7pm-10pm and 1am-6am weekdays, from 6pm-10pm Saturdays and weekend overnights; otherwise, they're pretty much local.
Same could be said for UKRD's takeover of the former TLRC [The Local Radio Company] stations; they sound like live and local stations of the 1990s, with emphasis on the local part, despite local radio being Deader Than Disco now. Plus, there were few if any rebrands, except in the North East, and no networking - so it feels local in content terms.
Pretty much all of the myriad conservative talk radio shows in the United States are spiritual successors to Rush Limbaugh. His was the first nationally syndicated conservative talk show to take advantage of the lifting of the Fairness Doctrine, which mandated that equal time be given to opposing viewpoints. In practice, the Fairness Doctrine made it unattractive for stations to host any kind of opinion show at all. After its lifting, stations were able to choose to air an opinion show without having to air a rebuttal of equal time, and Limbaugh was the first to take full advantage of this. YMMV on whether this is a good thing or not, but keep in mind that it also allowed for liberals to attempt to emulate the format for their views to varying degrees of success. Also, very theatrical angry types such as Michael Savage are spiritual successors to Morton Downey Jr.
Gloomsbury to The Wordsmiths at Gorsemere: a BBC Radio 4 slice-of-life sitcom written by Sue Limb about an exaggerated version of a well known group of writers (the Romantic poets and the Bloomsbury set), filled with punny names.