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There's, as you say, "BBC Radio 4",which is the national speech channel (NPR being the nearest American equivalent) - in a sense, this was the original "BBC" dating back to 1922, but they gave it "4" in the 1960s.

There's Channel 4, which is a non-profit national TV channel (though funded with ads, while the BBC channels are funded from the fixed licence fee), started in the 1980s, with a remit to be "alternative" to what existed then (which was 2 BBC channels - BBC One and BBC Two - and ITV, the for-profit channel started in the 1950s. If Channel 4 make a profit, it goes into the "Film 4" arm, which has made many of the best-known British films in the last few decades.

There's BBC Four, a TV channel started around 2000 (when digital TV allowed many more channels). This is a niche, low-budget channel, only starting at 7pm, with lots of documentaries, foreign films/TV series and comedies they're trying out. Plus lots of repeats.

(Plus there is also S4C - a Welsh language channel - Emrys would be the DUer to tell you about this)