In fact, Cat, I recall an election of several years ago (can't recall how far back) and all outlets had their correspondents dotted around the usual hot spots (Westminster, Number Ten, Leader's Constituencies, Trafalgar Square, wherever). I specifically recall that amongst the Lawleys, Dimbleby's, Paxman and Warks on the BBC, they had opted for none other than Esther Rantzen to be their correspondent outside Number Ten!!
Esther actually did a couple of elections for the Beeb, in 1974 she was "In The City" and interviewed various cleaners and businessmen about the state of the nation, in 1983 she was in Downing Street as you say, and in 1987 she was in Piccadilly Circus interviewing the Great British Public, as was Gavin Campbell. And of course her husband Demond Wilcox was in Traflagar Square for most of the seventies elections doing the time-honoured Bit Of Fun.
At the time, That's Life was actually made by the current affairs department - but that's because it was moved from features as Desmond Wilcox was Head of Features and there was a massive row over him favouring That's Life over everything else, which came to such a head that Wilcox quit and That's Life was moved to a different department to avoid ill-feeling.